Theorem axc11nlemOLD 2160

Description: Obsolete proof of axc11nlemOLD2 1988 as of 14-Mar-2021. (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.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
axc11nlemOLD.1	|- ( -. A. y y = x -> ( x = z -> A. y x = z ) )

Assertion

Ref	Expression
axc11nlemOLD	|- ( A. x x = z -> A. y y = x )

Distinct variable groups:   x, z   y, z

Proof of Theorem axc11nlemOLD

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		axc11nlemOLD.1	. . . . 5 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
7	6	spsd 2057	. . . 4 |- ( -. A. y y = x -> ( A. x x = z -> A. y x = z ) )
8	7	com12 32	. . 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  ax-12 2047

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

This theorem is referenced by: (None)