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Metamath Proof Explorer

Theorem hbae

Description: All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbaev when possible. (Contributed by NM, 13-May-1993) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

Ref Expression
Assertion hbae ⊢ ∀ x x = y → ∀ z ∀ x x = y

Proof

Step Hyp Ref Expression
1 sp ⊢ ∀ x x = y → x = y
2 axc9 ⊢ ¬ ∀ z z = x → ¬ ∀ z z = y → x = y → ∀ z x = y
3 1 2 syl7 ⊢ ¬ ∀ z z = x → ¬ ∀ z z = y → ∀ x x = y → ∀ z x = y
4 axc11r ⊢ ∀ z z = x → ∀ x x = y → ∀ z x = y
5 axc11 ⊢ ∀ x x = y → ∀ x x = y → ∀ y x = y
6 5 pm2.43i ⊢ ∀ x x = y → ∀ y x = y
7 axc11r ⊢ ∀ z z = y → ∀ y x = y → ∀ z x = y
8 6 7 syl5 ⊢ ∀ z z = y → ∀ x x = y → ∀ z x = y
9 3 4 8 pm2.61ii ⊢ ∀ x x = y → ∀ z x = y
10 9 axc4i ⊢ ∀ x x = y → ∀ x ∀ z x = y
11 ax-11 ⊢ ∀ x ∀ z x = y → ∀ z ∀ x x = y
12 10 11 syl ⊢ ∀ x x = y → ∀ z ∀ x x = y