Theorem rexprg 3635

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 3590	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	rexeqi 2670	. . 3 |- ( E. x e. { A , B } ph <-> E. x e. ( { A } u. { B } ) ph )
3		rexun 3307	. . 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 3624	. . . 4 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
7	6	orbi1d 786	. . 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 3624	. . . 4 |- ( B e. W -> ( E. x e. { B } ph <-> ch ) )
10	9	orbi2d 785	. . 3 |- ( B e. W -> ( ( ps \/ E. x e. { B } ph ) <-> ( ps \/ ch ) ) )
11	7, 10	sylan9bb 459	. 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 703   = wceq 1348   e. wcel 2141   E.wrex 2449   u. cun 3119   {csn 3583   {cpr 3584

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  rextpg  3637  rexpr  3639  minmax  11193  xrminmax  11228