Theorem a12stdy2 1415

Description: Part of a study related to ax-12 1009. The consequent is quantified with a different variable. There are no distinct variable restrictions.

Assertion

Ref	Expression
a12stdy2	|- (A.z(z = x /\ x = y) -> A.y y = x)

Proof of Theorem a12stdy2

Step	Hyp	Ref	Expression
1		19.26 1108	. 2 |- (A.z(z = x /\ x = y) <-> (A.z z = x /\ A.z x = y))
2		ax-10o 1182	. . . 4 |- (A.z z = x -> (A.z x = y -> A.x x = y))
3		alequcom 1184	. . . 4 |- (A.x x = y -> A.y y = x)
4	2, 3	syl6 22	. . 3 |- (A.z z = x -> (A.z x = y -> A.y y = x))
5	4	imp 357	. 2 |- ((A.z z = x /\ A.z x = y) -> A.y y = x)
6	1, 5	sylbi 206	1 |- (A.z(z = x /\ x = y) -> A.y y = x)

Syntax hints:   -> wi 3   /\ wa 230  A.wal 995   = wceq 997

This theorem is referenced by:  a12stdy3 1416

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-10 1007  ax-4 1014  ax-5o 1016  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-an 232

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