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Metamath Proof Explorer

Theorem hbae-o

Description: All variables are effectively bound in an identical variable specifier. Version of hbae using ax-c11 . (Contributed by NM, 13-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 ax-c5 ⊢ ∀ x x = y → x = y
2 ax-c9 ⊢ ¬ ∀ 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 ax-c11 ⊢ ∀ x x = z → ∀ x x = y → ∀ z x = y
5 4 aecoms-o ⊢ ∀ z z = x → ∀ x x = y → ∀ z x = y
6 ax-c11 ⊢ ∀ x x = y → ∀ x x = y → ∀ y x = y
7 6 pm2.43i ⊢ ∀ x x = y → ∀ y x = y
8 ax-c11 ⊢ ∀ y y = z → ∀ y x = y → ∀ z x = y
9 7 8 syl5 ⊢ ∀ y y = z → ∀ x x = y → ∀ z x = y
10 9 aecoms-o ⊢ ∀ z z = y → ∀ x x = y → ∀ z x = y
11 3 5 10 pm2.61ii ⊢ ∀ x x = y → ∀ z x = y
12 11 axc4i-o ⊢ ∀ x x = y → ∀ x ∀ z x = y
13 ax-11 ⊢ ∀ x ∀ z x = y → ∀ z ∀ x x = y
14 12 13 syl ⊢ ∀ x x = y → ∀ z ∀ x x = y