Theorem prtlem13 34153

Description: Lemma for prter1 34164, prter2 34166, prter3 34167 and prtex 34165. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem13.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }

Assertion

Ref	Expression
prtlem13	|- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )

Distinct variable groups:   v, u, x, y, A   w, v, x, y   z, v, x, y

Allowed substitution hints:   A( z, w)   .~ ( x, y, z, w, v, u)

Proof of Theorem prtlem13

Step	Hyp	Ref	Expression
1		vex 3203	. 2 |- z e. _V
2		vex 3203	. 2 |- w e. _V
3		elequ2 2004	. . . . 5 |- ( u = v -> ( x e. u <-> x e. v ) )
4		elequ2 2004	. . . . 5 |- ( u = v -> ( y e. u <-> y e. v ) )
5	3, 4	anbi12d 747	. . . 4 |- ( u = v -> ( ( x e. u /\ y e. u ) <-> ( x e. v /\ y e. v ) ) )
6	5	cbvrexv 3172	. . 3 |- ( E. u e. A ( x e. u /\ y e. u ) <-> E. v e. A ( x e. v /\ y e. v ) )
7		eleq1 2689	. . . . 5 |- ( x = z -> ( x e. v <-> z e. v ) )
8		eleq1 2689	. . . . 5 |- ( y = w -> ( y e. v <-> w e. v ) )
9	7, 8	bi2anan9 917	. . . 4 |- ( ( x = z /\ y = w ) -> ( ( x e. v /\ y e. v ) <-> ( z e. v /\ w e. v ) ) )
10	9	rexbidv 3052	. . 3 |- ( ( x = z /\ y = w ) -> ( E. v e. A ( x e. v /\ y e. v ) <-> E. v e. A ( z e. v /\ w e. v ) ) )
11	6, 10	syl5bb 272	. 2 |- ( ( x = z /\ y = w ) -> ( E. u e. A ( x e. u /\ y e. u ) <-> E. v e. A ( z e. v /\ w e. v ) ) )
12		prtlem13.1	. 2 |- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
13	1, 2, 11, 12	braba 4992	1 |- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wrex 2913   class class class wbr 4653   {copab 4712

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  prtlem16  34154  prtlem18  34162  prter1  34164  prter3  34167