Theorem a12stdy2 1371

Description: Part of a study related to ax-12 966. The consequent is quantified with a different variable. There are no distinct variable restrictions.

Assertion

Ref	Expression
a12stdy2	⊢ (∀z(z = x ⋀ x = y) → ∀y y = x)

Proof of Theorem a12stdy2

Step	Hyp	Ref	Expression
1		19.26 1065	. 2 ⊢ (∀z(z = x ⋀ x = y) ↔ (∀z z = x ⋀ ∀z x = y))
2		ax-10o 1138	. . . 4 ⊢ (∀z z = x → (∀z x = y → ∀x x = y))
3		alequcom 1140	. . . 4 ⊢ (∀x x = y → ∀y y = x)
4	2, 3	syl6 22	. . 3 ⊢ (∀z z = x → (∀z x = y → ∀y y = x))
5	4	imp 350	. 2 ⊢ ((∀z z = x ⋀ ∀z x = y) → ∀y y = x)
6	1, 5	sylbi 199	1 ⊢ (∀z(z = x ⋀ x = y) → ∀y y = x)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954

This theorem is referenced by:  a12stdy3 1372

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-10 964  ax-4 971  ax-5o 973  ax-10o 1138

This theorem depends on definitions:  df-bi 147  df-an 225

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