Theorem ralprg 3421

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 3382	. . . 4 |- { A , B } = ( { A } u. { B } )
2	1	raleqi 2509	. . 3 |- ( A. x e. { A , B } ph <-> A. x e. ( { A } u. { B } ) ph )
3		ralunb 3124	. . 3 |- ( A. x e. ( { A } u. { B } ) ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
4	2, 3	bitri 173	. 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 3411	. . 3 |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )
7		ralprg.2	. . . 4 |- ( x = B -> ( ph <-> ch ) )
8	7	ralsng 3411	. . 3 |- ( B e. W -> ( A. x e. { B } ph <-> ch ) )
9	6, 8	bi2anan9 538	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A. x e. { A } ph /\ A. x e. { B } ph ) <-> ( ps /\ ch ) ) )
10	4, 9	syl5bb 181	1 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   u. cun 2915   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  raltpg  3423  ralpr  3425  iinxprg  3731  fvinim0ffz  9096