Theorem bj-axc11nv 31392

Description: Version of axc11n 2153 with a dv condition, which does not require ax-13 2101. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc11nv	|- ( A. x x = y -> A. y y = x )

Distinct variable group:   x, y

Proof of Theorem bj-axc11nv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equcomi 1871	. . . . 5 |- ( z = x -> x = z )
2		ax-5 1768	. . . . . 6 |- ( x = z -> A. y x = z )
3	2	a1i 11	. . . . 5 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
4	1, 3	syl5com 31	. . . 4 |- ( z = x -> ( -. A. y y = x -> A. y x = z ) )
5		axc112 2030	. . . . 5 |- ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
6		bj-axc11nlemv 31391	. . . . 5 |- ( A. x x = z -> A. y y = x )
7	5, 6	syl6 34	. . . 4 |- ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
8	4, 7	syl9 73	. . 3 |- ( z = x -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
9		ax6ev 1817	. . 3 |- E. z z = x
10	8, 9	exlimiiv 1787	. 2 |- ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
11	10	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-12 1943

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

This theorem is referenced by:  bj-aecomsv  31393  bj-naecomsv  31394