Theorem axc11nALT 2154

Description: Alternate proof of axc11n 2153 from axc11nlemALT 2152. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11nALT	|- ( A. x x = y -> A. y y = x )

Proof of Theorem axc11nALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcomi 1871	. . . . 5 |- ( z = x -> x = z )
2		dveeq1 2148	. . . . 5 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
3	1, 2	syl5com 31	. . . 4 |- ( z = x -> ( -. A. y y = x -> A. y x = z ) )
4		axc112 2030	. . . . 5 |- ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
5		axc11nlemALT 2152	. . . . 5 |- ( A. x x = z -> A. y y = x )
6	4, 5	syl6 34	. . . 4 |- ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
7	3, 6	syl9 73	. . 3 |- ( z = x -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
8		ax6ev 1817	. . 3 |- E. z z = x
9	7, 8	exlimiiv 1787	. 2 |- ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
10	9	pm2.18d 116	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678

This theorem is referenced by: (None)