Theorem rexprg 3698

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 3653	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	rexeqi 2713	. . 3 |- ( E. x e. { A , B } ph <-> E. x e. ( { A } u. { B } ) ph )
3		rexun 3364	. . 3 |- ( E. x e. ( { A } u. { B } ) ph <-> ( E. x e. { A } ph \/ E. x e. { B } ph ) )
4	2, 3	bitri 184	. 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 3687	. . . 4 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
7	6	orbi1d 795	. . 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 3687	. . . 4 |- ( B e. W -> ( E. x e. { B } ph <-> ch ) )
10	9	orbi2d 794	. . 3 |- ( B e. W -> ( ( ps \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
11	7, 10	sylan9bb 462	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( E. x e. { A } ph \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
12	4, 11	bitrid 192	1 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 712   = wceq 1375   e. wcel 2180   E.wrex 2489   u. cun 3175   {csn 3646   {cpr 3647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-sn 3652  df-pr 3653

This theorem is referenced by:  rextpg  3700  rexpr  3702  minmax  11707  xrminmax  11742