Theorem ralprg 3645

Description: Convert a quantification over a pair to a conjunction. (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
ralprg	|- ( ( A e. V /\ B e. W ) -> ( A. 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 ralprg

Step	Hyp	Ref	Expression
1		df-pr 3601	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	raleqi 2677	. . 3 |- ( A. x e. { A , B } ph <-> A. x e. ( { A } u. { B } ) ph )
3		ralunb 3318	. . 3 |- ( A. x e. ( { A } u. { B } ) ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
4	2, 3	bitri 184	. 2 |- ( A. x e. { A , B } ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
5		ralprg.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	5	ralsng 3634	. . 3 |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )
7		ralprg.2	. . . 4 |- ( x = B -> ( ph <-> ch ) )
8	7	ralsng 3634	. . 3 |- ( B e. W -> ( A. x e. { B } ph <-> ch ) )
9	6, 8	bi2anan9 606	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A. x e. { A } ph /\ A. x e. { B } ph ) <-> ( ps /\ ch ) ) )
10	4, 9	bitrid 192	1 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   u. cun 3129   {csn 3594   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  raltpg  3647  ralpr  3649  iinxprg  3963  fvinim0ffz  10243  sumpr  11423