Theorem axc11nlemOLD2 1988

Description: Lemma for axc11n 2307. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1983 instead. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
axc11nlemOLD2.1	|- ( -. A. y y = x -> ( x = z -> A. y x = z ) )

Assertion

Ref	Expression
axc11nlemOLD2	|- ( A. x x = z -> A. y y = x )

Distinct variable groups:   x, z   y, z

Proof of Theorem axc11nlemOLD2

Step	Hyp	Ref	Expression
1		cbvaev 1979	. . 3 |- ( A. x x = z -> A. y y = z )
2		equequ2 1953	. . . . 5 |- ( x = z -> ( y = x <-> y = z ) )
3	2	biimprd 238	. . . 4 |- ( x = z -> ( y = z -> y = x ) )
4	3	al2imi 1743	. . 3 |- ( A. y x = z -> ( A. y y = z -> A. y y = x ) )
5	1, 4	syl5com 31	. 2 |- ( A. x x = z -> ( A. y x = z -> A. y y = x ) )
6		spaev 1978	. . . 4 |- ( A. x x = z -> x = z )
7		axc11nlemOLD2.1	. . . 4 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
8	6, 7	syl5com 31	. . 3 |- ( A. x x = z -> ( -. A. y y = x -> A. y x = z ) )
9	8	con1d 139	. 2 |- ( A. x x = z -> ( -. A. y x = z -> A. y y = x ) )
10	5, 9	pm2.61d 170	1 |- ( A. x x = z -> A. y y = x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  aevlemOLD  1989  axc11nOLDOLD  2309