Theorem prtlem18 34162

Description: Lemma for prter2 34166. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }

Assertion

Ref	Expression
prtlem18	|- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v <-> z .~ w ) ) )

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

Allowed substitution hints:   .~ ( x, y, u)

Proof of Theorem prtlem18

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 3003	. . . . 5 |- ( ( v e. A /\ ( z e. v /\ w e. v ) ) -> E. v e. A ( z e. v /\ w e. v ) )
2	1	expr 643	. . . 4 |- ( ( v e. A /\ z e. v ) -> ( w e. v -> E. v e. A ( z e. v /\ w e. v ) ) )
3		prtlem18.1	. . . . 5 |- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
4	3	prtlem13 34153	. . . 4 |- ( z .~ w <-> E. v e. A ( z e. v /\ w e. v ) )
5	2, 4	syl6ibr 242	. . 3 |- ( ( v e. A /\ z e. v ) -> ( w e. v -> z .~ w ) )
6	5	a1i 11	. 2 |- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v -> z .~ w ) ) )
7	3	prtlem13 34153	. . 3 |- ( z .~ w <-> E. p e. A ( z e. p /\ w e. p ) )
8		prtlem17 34161	. . 3 |- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( E. p e. A ( z e. p /\ w e. p ) -> w e. v ) ) )
9	7, 8	syl7bi 245	. 2 |- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( z .~ w -> w e. v ) ) )
10	6, 9	impbidd 200	1 |- ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v <-> z .~ w ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   {copab 4712   Prt wprt 34156

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  df-prt 34157

This theorem is referenced by:  prtlem19  34163