Theorem ax6e2eq 32285

Description: Alternate form of ax6e 1964 for non-distinct x, y and u = v. ax6e2eq 32285 is derived from ax6e2eqVD 32662. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2eq	|- ( A. x x = y -> ( u = v -> E. x E. y ( x = u /\ y = v ) ) )

Distinct variable groups:   x, u   y, u   x, v   y, v

Proof of Theorem ax6e2eq

Step	Hyp	Ref	Expression
1		ax6ev 1716	. . . . . . 7 |- E. x x = u
2		hbae 2021	. . . . . . . 8 |- ( A. x x = y -> A. x A. x x = y )
3		ax-7 1734	. . . . . . . . . 10 |- ( x = y -> ( x = u -> y = u ) )
4	3	sps 1809	. . . . . . . . 9 |- ( A. x x = y -> ( x = u -> y = u ) )
5	4	ancld 553	. . . . . . . 8 |- ( A. x x = y -> ( x = u -> ( x = u /\ y = u ) ) )
6	2, 5	eximdh 1645	. . . . . . 7 |- ( A. x x = y -> ( E. x x = u -> E. x ( x = u /\ y = u ) ) )
7	1, 6	mpi 17	. . . . . 6 |- ( A. x x = y -> E. x ( x = u /\ y = u ) )
8	7	axc4i 1841	. . . . 5 |- ( A. x x = y -> A. x E. x ( x = u /\ y = u ) )
9		axc11 2020	. . . . 5 |- ( A. x x = y -> ( A. x E. x ( x = u /\ y = u ) -> A. y E. x ( x = u /\ y = u ) ) )
10	8, 9	mpd 15	. . . 4 |- ( A. x x = y -> A. y E. x ( x = u /\ y = u ) )
11		19.2 1718	. . . 4 |- ( A. y E. x ( x = u /\ y = u ) -> E. y E. x ( x = u /\ y = u ) )
12	10, 11	syl 16	. . 3 |- ( A. x x = y -> E. y E. x ( x = u /\ y = u ) )
13		excomim 1794	. . 3 |- ( E. y E. x ( x = u /\ y = u ) -> E. x E. y ( x = u /\ y = u ) )
14	12, 13	syl 16	. 2 |- ( A. x x = y -> E. x E. y ( x = u /\ y = u ) )
15		equtrr 1741	. . . 4 |- ( u = v -> ( y = u -> y = v ) )
16	15	anim2d 565	. . 3 |- ( u = v -> ( ( x = u /\ y = u ) -> ( x = u /\ y = v ) ) )
17	16	2eximdv 1683	. 2 |- ( u = v -> ( E. x E. y ( x = u /\ y = u ) -> E. x E. y ( x = u /\ y = v ) ) )
18	14, 17	syl5com 30	1 |- ( A. x x = y -> ( u = v -> E. x E. y ( x = u /\ y = v ) ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1372   E.wex 1591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595

This theorem is referenced by:  ax6e2ndeq  32287  ax6e2ndeqVD  32664  ax6e2ndeqALT  32686