Theorem rexprg 4235

Description: Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralprg.1	|- ( x = A -> ( ph <-> ps ) )
ralprg.2	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
rexprg	|- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )

Distinct variable groups:   x, A   x, B   ps, x   ch, x

Allowed substitution hints:   ph( x)   V( x)   W( x)

Proof of Theorem rexprg

Step	Hyp	Ref	Expression
1		df-pr 4180	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	rexeqi 3143	. . 3 |- ( E. x e. { A , B } ph <-> E. x e. ( { A } u. { B } ) ph )
3		rexun 3793	. . 3 |- ( E. x e. ( { A } u. { B } ) ph <-> ( E. x e. { A } ph \/ E. x e. { B } ph ) )
4	2, 3	bitri 264	. 2 |- ( E. x e. { A , B } ph <-> ( E. x e. { A } ph \/ E. x e. { B } ph ) )
5		ralprg.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	rexsng 4219	. . . 4 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
7	6	orbi1d 739	. . 3 |- ( A e. V -> ( ( E. x e. { A } ph \/ E. x e. { B } ph ) <-> ( ps \/ E. x e. { B } ph ) ) )
8		ralprg.2	. . . . 5 |- ( x = B -> ( ph <-> ch ) )
9	8	rexsng 4219	. . . 4 |- ( B e. W -> ( E. x e. { B } ph <-> ch ) )
10	9	orbi2d 738	. . 3 |- ( B e. W -> ( ( ps \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
11	7, 10	sylan9bb 736	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( E. x e. { A } ph \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
12	4, 11	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   {csn 4177   {cpr 4179

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  rextpg  4237  rexpr  4239  fr2nr  5092  sgrp2nmndlem5  17416  nb3grprlem2  26283  nfrgr2v  27136  3vfriswmgrlem  27141  brfvrcld  37983  rnmptpr  39358  ldepspr  42262  zlmodzxzldeplem4  42292