Theorem a12stdy1 1414

Description: Part of a study related to ax-12 1009. The consequent introduces a new variable z. There are no distinct variable restrictions.

Assertion

Ref	Expression
a12stdy1	|- (A.x x = y -> E.x y = z)

Proof of Theorem a12stdy1

Step	Hyp	Ref	Expression
1		a9e 1166	. 2 |- E.y y = z
2		ax-10o 1182	. . . 4 |- (A.x x = y -> (A.x -. y = z -> A.y -. y = z))
3	2	con3d 99	. . 3 |- (A.x x = y -> (-. A.y -. y = z -> -. A.x -. y = z))
4		df-ex 1022	. . 3 |- (E.y y = z <-> -. A.y -. y = z)
5		df-ex 1022	. . 3 |- (E.x y = z <-> -. A.x -. y = z)
6	3, 4, 5	3imtr4g 564	. 2 |- (A.x x = y -> (E.y y = z -> E.x y = z))
7	1, 6	mpi 44	1 |- (A.x x = y -> E.x y = z)

Syntax hints:  -. wn 2   -> wi 3  A.wal 995   = wceq 997  E.wex 1021

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-9 1006  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022

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