Theorem a12stdy4 1417

Description: Part of a study related to ax-12 1009. The second antecedent of ax-12 1009 is replaced. There are no distinct variable restrictions.

Assertion

Ref	Expression
a12stdy4	|- (-. A.z z = x -> (A.y z = x -> (x = y -> A.z x = y)))

Proof of Theorem a12stdy4

Step	Hyp	Ref	Expression
1		ax-10o 1182	. . . . . . 7 |- (A.y y = z -> (A.y z = x -> A.z z = x))
2	1	alequcoms 1185	. . . . . 6 |- (A.z z = y -> (A.y z = x -> A.z z = x))
3	2	con3d 99	. . . . 5 |- (A.z z = y -> (-. A.z z = x -> -. A.y z = x))
4	3	impcom 358	. . . 4 |- ((-. A.z z = x /\ A.z z = y) -> -. A.y z = x)
5	4	pm2.21d 81	. . 3 |- ((-. A.z z = x /\ A.z z = y) -> (A.y z = x -> (x = y -> A.z x = y)))
6	5	ex 380	. 2 |- (-. A.z z = x -> (A.z z = y -> (A.y z = x -> (x = y -> A.z x = y))))
7		ax-12 1009	. . 3 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
8	7	a1dd 42	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (A.y z = x -> (x = y -> A.z x = y))))
9	6, 8	pm2.61d 133	1 |- (-. A.z z = x -> (A.y z = x -> (x = y -> A.z x = y)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 230  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-10 1007  ax-12 1009  ax-10o 1182

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

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