diff --git a/library/init/funext.lean b/library/init/funext.lean
index 454263bc6..761feefe9 100644
--- a/library/init/funext.lean
+++ b/library/init/funext.lean
@@ -56,7 +56,7 @@ section
     ... = f₂              : rfl
 end
 
-attribute funext [intro]
+attribute funext [intro!]
 
 open function.equiv_notation
 
diff --git a/library/init/logic.lean b/library/init/logic.lean
index 2ad5df110..4426285bf 100644
--- a/library/init/logic.lean
+++ b/library/init/logic.lean
@@ -24,7 +24,7 @@ prefix `¬` := not
 definition absurd {a : Prop} {b : Type} (H1 : a) (H2 : ¬a) : b :=
 false.rec b (H2 H1)
 
-lemma not.intro [intro] {a : Prop} (H : a → false) : ¬ a :=
+lemma not.intro [intro!] {a : Prop} (H : a → false) : ¬ a :=
 H
 
 theorem mt {a b : Prop} (H1 : a → b) (H2 : ¬b) : ¬a :=
@@ -243,7 +243,7 @@ notation a ∧ b  := and a b
 variables {a b c d : Prop}
 
 attribute and.rec [elim]
-attribute and.intro [intro]
+attribute and.intro [intro!]
 
 theorem and.elim (H₁ : a ∧ b) (H₂ : a → b → c) : c :=
 and.rec H₂ H₁
@@ -280,7 +280,7 @@ notation a ↔ b := iff a b
 
 theorem iff.intro : (a → b) → (b → a) → (a ↔ b) := and.intro
 
-attribute iff.intro [intro]
+attribute iff.intro [intro!]
 
 theorem iff.elim : ((a → b) → (b → a) → c) → (a ↔ b) → c := and.rec
 
@@ -508,9 +508,7 @@ and_congr (imp_congr H1 H2) (imp_congr H2 H1)
 inductive Exists {A : Type} (P : A → Prop) : Prop :=
 intro : ∀ (a : A), P a → Exists P
 
--- Remark: we don't mark exists.intro with [intro] because the [intro]
--- rules are applied eagerly.
-attribute Exists.intro [backward]
+attribute Exists.intro [intro]
 
 definition exists.intro := @Exists.intro
 
@@ -530,7 +528,7 @@ definition exists_unique {A : Type} (p : A → Prop) :=
 
 notation `∃!` binders `, ` r:(scoped P, exists_unique P) := r
 
-theorem exists_unique.intro [backward] {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
+theorem exists_unique.intro [intro] {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
   ∃!x, p x :=
 exists.intro w (and.intro H1 H2)
 
diff --git a/src/emacs/lean-syntax.el b/src/emacs/lean-syntax.el
index 7555be368..2a5e56996 100644
--- a/src/emacs/lean-syntax.el
+++ b/src/emacs/lean-syntax.el
@@ -55,9 +55,9 @@
            "class" "parsing_only" "coercion" "unfold_full" "constructor"
            "reducible" "irreducible" "semireducible" "quasireducible" "wf"
            "whnf" "multiple_instances" "none" "decls" "declarations"
-           "coercions" "classes" "symm" "subst" "refl" "trans" "simp" "simps" "congr" "backward"
+           "coercions" "classes" "symm" "subst" "refl" "trans" "simp" "simps" "congr"
            "forward" "no_pattern" "notations" "abbreviations" "begin_end_hints" "tactic_hints"
-           "reduce_hints" "unfold_hints" "aliases" "eqv" "intro" "elim"
+           "reduce_hints" "unfold_hints" "aliases" "eqv" "intro" "intro!" "elim"
            "localrefinfo" "recursor"))
   "lean modifiers")
 (defconst lean-modifiers-regexp
diff --git a/src/frontends/lean/builtin_cmds.cpp b/src/frontends/lean/builtin_cmds.cpp
index f2e2df9c5..a1716cb82 100644
--- a/src/frontends/lean/builtin_cmds.cpp
+++ b/src/frontends/lean/builtin_cmds.cpp
@@ -709,7 +709,7 @@ environment print_cmd(parser & p) {
     } else if (p.curr_is_token(get_simp_attr_tk())) {
         p.next();
         print_simp_rules(p);
-    } else if (p.curr_is_token(get_intro_attr_tk())) {
+    } else if (p.curr_is_token(get_intro_bang_attr_tk())) {
         p.next();
         print_intro_lemmas(p);
     } else if (p.curr_is_token(get_elim_attr_tk())) {
@@ -722,7 +722,7 @@ environment print_cmd(parser & p) {
         p.next();
         p.check_token_next(get_rbracket_tk(), "invalid 'print [light]', ']' expected");
         print_light_rules(p);
-    } else if (p.curr_is_token(get_backward_attr_tk())) {
+    } else if (p.curr_is_token(get_intro_attr_tk())) {
         p.next();
         print_backward_rules(p);
     } else if (print_polymorphic(p)) {
diff --git a/src/frontends/lean/decl_attributes.cpp b/src/frontends/lean/decl_attributes.cpp
index b3e887e86..b76d79997 100644
--- a/src/frontends/lean/decl_attributes.cpp
+++ b/src/frontends/lean/decl_attributes.cpp
@@ -46,9 +46,9 @@ decl_attributes::decl_attributes(bool is_abbrev, bool persistent):
     m_recursor               = false;
     m_simp                   = false;
     m_congr                  = false;
-    m_backward               = false;
     m_forward                = false;
     m_intro                  = false;
+    m_intro_bang             = false;
     m_elim                   = false;
     m_no_pattern             = false;
 }
@@ -97,9 +97,9 @@ void decl_attributes::parse(parser & p) {
             p.next();
         } else if (auto it = parse_priority(p)) {
             m_priority = *it;
-            if (!m_is_instance && !m_simp && !m_congr && !m_backward && !m_forward && !m_elim && !m_intro) {
+            if (!m_is_instance && !m_simp && !m_congr && !m_forward && !m_elim && !m_intro && !m_intro_bang) {
                 throw parser_error("invalid '[priority]' attribute, declaration must be marked as an '[instance]', '[simp]', '[congr]', "
-                                   "'[backward]', '[forward]', '[intro]' or '[elim]'", pos);
+                                   "'[forward]', '[intro!]', '[intro]' or '[elim]'", pos);
             }
         } else if (p.curr_is_token(get_parsing_only_tk())) {
             if (!m_is_abbrev)
@@ -148,9 +148,6 @@ void decl_attributes::parse(parser & p) {
             p.next();
             m_light_arg = p.parse_small_nat();
             p.check_token_next(get_rbracket_tk(), "light attribute has form '[light <i>]'");
-        } else if (p.curr_is_token(get_backward_attr_tk())) {
-            p.next();
-            m_backward = true;
         } else if (p.curr_is_token(get_forward_attr_tk())) {
             p.next();
             m_forward = true;
@@ -160,6 +157,9 @@ void decl_attributes::parse(parser & p) {
         } else if (p.curr_is_token(get_intro_attr_tk())) {
             p.next();
             m_intro = true;
+        } else if (p.curr_is_token(get_intro_bang_attr_tk())) {
+            p.next();
+            m_intro_bang = true;
         } else if (p.curr_is_token(get_no_pattern_attr_tk())) {
             p.next();
             m_no_pattern = true;
@@ -252,12 +252,6 @@ environment decl_attributes::apply(environment env, io_state const & ios, name c
     if (m_light_arg) {
         env = add_light_rule(env, d, *m_light_arg, ns, m_persistent);
     }
-    if (m_backward) {
-        if (m_priority)
-            env = add_backward_rule(env, d, *m_priority, ns, m_persistent);
-        else
-            env = add_backward_rule(env, d, LEAN_BACKWARD_DEFAULT_PRIORITY, ns, m_persistent);
-    }
     if (forward) {
         mk_multipatterns(env, ios, d); // try to create patterns
         if (m_priority)
@@ -266,6 +260,12 @@ environment decl_attributes::apply(environment env, io_state const & ios, name c
             env = add_forward_lemma(env, d, LEAN_FORWARD_DEFAULT_PRIORITY, ns, m_persistent);
     }
     if (m_intro) {
+        if (m_priority)
+            env = add_backward_rule(env, d, *m_priority, ns, m_persistent);
+        else
+            env = add_backward_rule(env, d, LEAN_BACKWARD_DEFAULT_PRIORITY, ns, m_persistent);
+    }
+    if (m_intro_bang) {
         if (m_priority)
             env = add_intro_lemma(env, ios, d, *m_priority, ns, m_persistent);
         else
@@ -290,8 +290,8 @@ void decl_attributes::write(serializer & s) const {
       << m_is_reducible << m_is_irreducible << m_is_semireducible << m_is_quasireducible
       << m_is_class << m_is_parsing_only << m_has_multiple_instances << m_unfold_full_hint
       << m_constructor_hint << m_symm << m_trans << m_refl << m_subst << m_recursor
-      << m_simp << m_congr << m_recursor_major_pos << m_priority << m_backward
-      << m_forward << m_elim << m_intro << m_no_pattern;
+      << m_simp << m_congr << m_recursor_major_pos << m_priority
+      << m_forward << m_elim << m_intro << m_intro_bang << m_no_pattern;
     write_list(s, m_unfold_hint);
 }
 
@@ -300,8 +300,8 @@ void decl_attributes::read(deserializer & d) {
       >> m_is_reducible >> m_is_irreducible >> m_is_semireducible >> m_is_quasireducible
       >> m_is_class >> m_is_parsing_only >> m_has_multiple_instances >> m_unfold_full_hint
       >> m_constructor_hint >> m_symm >> m_trans >> m_refl >> m_subst >> m_recursor
-      >> m_simp >> m_congr >> m_recursor_major_pos >> m_priority >> m_backward
-      >> m_forward >> m_elim >> m_intro >> m_no_pattern;
+      >> m_simp >> m_congr >> m_recursor_major_pos >> m_priority
+      >> m_forward >> m_elim >> m_intro >> m_intro_bang >> m_no_pattern;
     m_unfold_hint = read_list<unsigned>(d);
 }
 }
diff --git a/src/frontends/lean/decl_attributes.h b/src/frontends/lean/decl_attributes.h
index 1a5df1a48..963ce6f3c 100644
--- a/src/frontends/lean/decl_attributes.h
+++ b/src/frontends/lean/decl_attributes.h
@@ -31,9 +31,9 @@ class decl_attributes {
     bool               m_recursor;
     bool               m_simp;
     bool               m_congr;
-    bool               m_backward;
     bool               m_forward;
     bool               m_intro;
+    bool               m_intro_bang;
     bool               m_elim;
     bool               m_no_pattern;
     optional<unsigned> m_recursor_major_pos;
diff --git a/src/frontends/lean/token_table.cpp b/src/frontends/lean/token_table.cpp
index 36f1e0c9b..09f367ff5 100644
--- a/src/frontends/lean/token_table.cpp
+++ b/src/frontends/lean/token_table.cpp
@@ -111,8 +111,8 @@ void init_token_table(token_table & t) {
          "[persistent]", "[visible]", "[instance]", "[trans_instance]", "[light",
          "[none]", "[class]", "[coercion]", "[reducible]", "[irreducible]", "[semireducible]", "[quasireducible]",
          "[simp]", "[congr]", "[parsing_only]", "[multiple_instances]", "[symm]", "[trans]", "[refl]", "[subst]", "[recursor",
-         "evaluate", "check", "eval", "[wf]", "[whnf]", "[priority", "[unfold_full]", "[unfold_hints]", "[backward]",
-         "[forward]", "[no_pattern]", "[constructor]", "[elim]", "[intro]", "[unfold", "print",
+         "evaluate", "check", "eval", "[wf]", "[whnf]", "[priority", "[unfold_full]", "[unfold_hints]", "[intro]",
+         "[forward]", "[no_pattern]", "[constructor]", "[elim]", "[intro!]", "[unfold", "print",
          "end", "namespace", "section", "prelude", "help",
          "import", "inductive", "record", "structure", "module", "universe", "universes", "local",
          "precedence", "reserve", "infixl", "infixr", "infix", "postfix", "prefix", "notation",
diff --git a/src/frontends/lean/tokens.cpp b/src/frontends/lean/tokens.cpp
index e1f365a49..b00a376a3 100644
--- a/src/frontends/lean/tokens.cpp
+++ b/src/frontends/lean/tokens.cpp
@@ -121,10 +121,10 @@ static name const * g_subst_tk = nullptr;
 static name const * g_simp_attr_tk = nullptr;
 static name const * g_congr_attr_tk = nullptr;
 static name const * g_light_attr_tk = nullptr;
-static name const * g_backward_attr_tk = nullptr;
 static name const * g_no_pattern_attr_tk = nullptr;
 static name const * g_forward_attr_tk = nullptr;
 static name const * g_intro_attr_tk = nullptr;
+static name const * g_intro_bang_attr_tk = nullptr;
 static name const * g_elim_attr_tk = nullptr;
 static name const * g_recursor_tk = nullptr;
 static name const * g_attribute_tk = nullptr;
@@ -281,10 +281,10 @@ void initialize_tokens() {
     g_simp_attr_tk = new name{"[simp]"};
     g_congr_attr_tk = new name{"[congr]"};
     g_light_attr_tk = new name{"[light"};
-    g_backward_attr_tk = new name{"[backward]"};
     g_no_pattern_attr_tk = new name{"[no_pattern]"};
     g_forward_attr_tk = new name{"[forward]"};
     g_intro_attr_tk = new name{"[intro]"};
+    g_intro_bang_attr_tk = new name{"[intro!]"};
     g_elim_attr_tk = new name{"[elim]"};
     g_recursor_tk = new name{"[recursor"};
     g_attribute_tk = new name{"attribute"};
@@ -442,10 +442,10 @@ void finalize_tokens() {
     delete g_simp_attr_tk;
     delete g_congr_attr_tk;
     delete g_light_attr_tk;
-    delete g_backward_attr_tk;
     delete g_no_pattern_attr_tk;
     delete g_forward_attr_tk;
     delete g_intro_attr_tk;
+    delete g_intro_bang_attr_tk;
     delete g_elim_attr_tk;
     delete g_recursor_tk;
     delete g_attribute_tk;
@@ -602,10 +602,10 @@ name const & get_subst_tk() { return *g_subst_tk; }
 name const & get_simp_attr_tk() { return *g_simp_attr_tk; }
 name const & get_congr_attr_tk() { return *g_congr_attr_tk; }
 name const & get_light_attr_tk() { return *g_light_attr_tk; }
-name const & get_backward_attr_tk() { return *g_backward_attr_tk; }
 name const & get_no_pattern_attr_tk() { return *g_no_pattern_attr_tk; }
 name const & get_forward_attr_tk() { return *g_forward_attr_tk; }
 name const & get_intro_attr_tk() { return *g_intro_attr_tk; }
+name const & get_intro_bang_attr_tk() { return *g_intro_bang_attr_tk; }
 name const & get_elim_attr_tk() { return *g_elim_attr_tk; }
 name const & get_recursor_tk() { return *g_recursor_tk; }
 name const & get_attribute_tk() { return *g_attribute_tk; }
diff --git a/src/frontends/lean/tokens.h b/src/frontends/lean/tokens.h
index 761415e61..c2a506ef7 100644
--- a/src/frontends/lean/tokens.h
+++ b/src/frontends/lean/tokens.h
@@ -123,10 +123,10 @@ name const & get_subst_tk();
 name const & get_simp_attr_tk();
 name const & get_congr_attr_tk();
 name const & get_light_attr_tk();
-name const & get_backward_attr_tk();
 name const & get_no_pattern_attr_tk();
 name const & get_forward_attr_tk();
 name const & get_intro_attr_tk();
+name const & get_intro_bang_attr_tk();
 name const & get_elim_attr_tk();
 name const & get_recursor_tk();
 name const & get_attribute_tk();
diff --git a/src/frontends/lean/tokens.txt b/src/frontends/lean/tokens.txt
index cd1c85207..7736dc3d0 100644
--- a/src/frontends/lean/tokens.txt
+++ b/src/frontends/lean/tokens.txt
@@ -116,10 +116,10 @@ subst        [subst]
 simp_attr    [simp]
 congr_attr   [congr]
 light_attr   [light
-backward_attr   [backward]
 no_pattern_attr [no_pattern]
-forward_attr [forward]
-intro_attr   [intro]
+forward_attr    [forward]
+intro_attr      [intro]
+intro_bang_attr [intro!]
 elim_attr    [elim]
 recursor     [recursor
 attribute    attribute
diff --git a/tests/lean/backward_rule1.lean b/tests/lean/backward_rule1.lean
index e5e505dc9..e4841e5ea 100644
--- a/tests/lean/backward_rule1.lean
+++ b/tests/lean/backward_rule1.lean
@@ -1,11 +1,11 @@
 constants (A B C : Prop) (H : A → B) (G : A → B → C)
 constants (T : Type) (f : T → A)
-attribute H [backward]
-attribute G [backward]
-attribute f [backward]
+attribute H [intro]
+attribute G [intro]
+attribute f [intro]
 
 print H
 print G
 print f
 
-print [backward]
+print [intro]
diff --git a/tests/lean/run/blast18.lean b/tests/lean/run/blast18.lean
index 6bd84f2eb..53311c07b 100644
--- a/tests/lean/run/blast18.lean
+++ b/tests/lean/run/blast18.lean
@@ -2,15 +2,15 @@
 set_option blast.strategy "backward"
 constants {P Q R S T U : Prop} (Hpq : P → Q) (Hqr : Q → R) (Hrs : R → S) (Hst : S → T)
 constants (Huq : U → Q) (Hur : U → R) (Hus : U → S) (Hut : U → T)
-attribute Hpq [backward]
-attribute Hqr [backward]
-attribute Hrs [backward]
-attribute Hst [backward]
+attribute Hpq [intro]
+attribute Hqr [intro]
+attribute Hrs [intro]
+attribute Hst [intro]
 
-attribute Huq [backward]
-attribute Hur [backward]
-attribute Hus [backward]
-attribute Hut [backward]
+attribute Huq [intro]
+attribute Hur [intro]
+attribute Hus [intro]
+attribute Hut [intro]
 
 definition lemma1 (p : P) : Q := by blast
 definition lemma2 (p : P) : R := by blast
diff --git a/tests/lean/run/blast19.lean b/tests/lean/run/blast19.lean
index 870f16b7a..32ff2b8a6 100644
--- a/tests/lean/run/blast19.lean
+++ b/tests/lean/run/blast19.lean
@@ -2,10 +2,10 @@
 set_option blast.strategy "backward"
 constants {P Q R S T U : Prop}
 constants (Huq : U → Q) (Hur : U → R) (Hus : U → S) (Hut : U → T)
-attribute Huq [backward]
-attribute Hur [backward]
-attribute Hus [backward]
-attribute Hut [backward]
+attribute Huq [intro]
+attribute Hur [intro]
+attribute Hus [intro]
+attribute Hut [intro]
 
 definition lemma1 : (P → Q) → P → Q := by blast
 definition lemma2 : (P → Q) → (Q → R) → P → R := by blast
diff --git a/tests/lean/run/blast20.lean b/tests/lean/run/blast20.lean
index 362455e22..593ab750f 100644
--- a/tests/lean/run/blast20.lean
+++ b/tests/lean/run/blast20.lean
@@ -12,7 +12,7 @@ by blast
 example : ∀ (F F' : Prop), F ∧ F' → F :=
 by blast
 
-attribute and.intro [backward]
+attribute and.intro [intro]
 
 example : ∀ (P Q R : nat → Prop) (F : Prop), (F ∧ (∀ n m, (Q m ∧ R n) → P n)) →
             (F → R 2) → Q 1 → P 2 ∧ F :=
diff --git a/tests/lean/run/blast21.lean b/tests/lean/run/blast21.lean
index c897154a9..1def7095f 100644
--- a/tests/lean/run/blast21.lean
+++ b/tests/lean/run/blast21.lean
@@ -9,7 +9,7 @@ constant subtype_refl : ∀ T, subtype T T
 
 constant subtype_trans : ∀ S T U, subtype S T → subtype T U → subtype S U
 
-attribute subtype_refl subtype_trans [backward]
+attribute subtype_refl subtype_trans [intro]
 
 lemma L1 : ∀ T1 T2 T3 T4 T5 T6, subtype T1 T2 → subtype T2 T3 → subtype T3 T4 → subtype T4 T5 → subtype T5 T6 → subtype T1 T6 :=
 by blast
diff --git a/tests/lean/run/blast_grind1.lean b/tests/lean/run/blast_grind1.lean
index 05f47291a..cce566bb6 100644
--- a/tests/lean/run/blast_grind1.lean
+++ b/tests/lean/run/blast_grind1.lean
@@ -11,7 +11,7 @@ by blast
 
 print ex1
 
-attribute Exists.intro [intro]
+attribute Exists.intro [intro!] -- grind and core_grind only process [intro!] declarations
 
 example (p q : nat → Prop) : (∃ x, p x ∧ q x) → (∃ x, q x) ∧ (∃ x, p x) :=
 by blast
@@ -27,7 +27,5 @@ by blast
 example (a : nat) : a = 0 → (λ x, x + a) = (λ x, x + 0) :=
 by blast
 
-attribute Exists.intro [intro]
-
 example (p q : nat → Prop) : (∃ x, p x ∧ q x) → (∃ x, q x) ∧ (∃ x, p x) :=
 by blast