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Theorem prtlem13 30771
 Description: Lemma for prter1 30782, prter2 30784, prter3 30785 and prtex 30783. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1
Assertion
Ref Expression
prtlem13
Distinct variable groups:   ,,,,   ,,,   ,,,
Allowed substitution hints:   (,)   (,,,,,)

Proof of Theorem prtlem13
StepHypRef Expression
1 vex 3112 . 2
2 vex 3112 . 2
3 elequ2 1824 . . . . 5
4 elequ2 1824 . . . . 5
53, 4anbi12d 710 . . . 4
65cbvrexv 3085 . . 3
7 eleq1 2529 . . . . 5
8 eleq1 2529 . . . . 5
97, 8bi2anan9 873 . . . 4
109rexbidv 2968 . . 3
116, 10syl5bb 257 . 2
12 prtlem13.1 . 2
131, 2, 11, 12braba 4773 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1395  wrex 2808   class class class wbr 4456  copab 4514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516 This theorem is referenced by:  prtlem16  30772  prtlem18  30780  prter1  30782  prter3  30785
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