Theorem a12stdy2 1366

Description: Part of a study related to ax-12 965. 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 1063	. 2 |- (A.z(z = x /\ x = y) <-> (A.z z = x /\ A.z x = y))
2		ax-10o 1136	. . . 4 |- (A.z z = x -> (A.z x = y -> A.x x = y))
3		alequcom 1138	. . . 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 350	. 2 |- ((A.z z = x /\ A.z x = y) -> A.y y = x)
6	1, 5	sylbi 199	1 |- (A.z(z = x /\ x = y) -> A.y y = x)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953

This theorem is referenced by:  a12stdy3 1367

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-10 963  ax-4 970  ax-5o 972  ax-10o 1136

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

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