Theorem rexprg 3628

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 3583	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	rexeqi 2666	. . 3 |- ( E. x e. { A , B } ph <-> E. x e. ( { A } u. { B } ) ph )
3		rexun 3302	. . 3 |- ( E. x e. ( { A } u. { B } ) ph <-> ( E. x e. { A } ph \/ E. x e. { B } ph ) )
4	2, 3	bitri 183	. 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 3617	. . . 4 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
7	6	orbi1d 781	. . 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 3617	. . . 4 |- ( B e. W -> ( E. x e. { B } ph <-> ch ) )
10	9	orbi2d 780	. . 3 |- ( B e. W -> ( ( ps \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
11	7, 10	sylan9bb 458	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( E. x e. { A } ph \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
12	4, 11	syl5bb 191	1 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   E.wrex 2445   u. cun 3114   {csn 3576   {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  rextpg  3630  rexpr  3632  minmax  11171  xrminmax  11206