Theorem raltpg 3445

Description: Convert a quantification over a triple 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 ) )
raltpg.3	|- ( x = C -> ( ph <-> th ) )

Assertion

Ref	Expression
raltpg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )

Distinct variable groups:   x, A   x, B   x, C   ps, x   ch, x   th, x

Allowed substitution hints:   ph( x)   V( x)   W( x)   X( x)

Proof of Theorem raltpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
2		ralprg.2	. . . . 5 |- ( x = B -> ( ph <-> ch ) )
3	1, 2	ralprg 3443	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
4		raltpg.3	. . . . 5 |- ( x = C -> ( ph <-> th ) )
5	4	ralsng 3433	. . . 4 |- ( C e. X -> ( A. x e. { C } ph <-> th ) )
6	3, 5	bi2anan9 570	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( ( A. x e. { A , B } ph /\ A. x e. { C } ph ) <-> ( ( ps /\ ch ) /\ th ) ) )
7	6	3impa 1133	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A. x e. { A , B } ph /\ A. x e. { C } ph ) <-> ( ( ps /\ ch ) /\ th ) ) )
8		df-tp 3406	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
9	8	raleqi 2553	. . 3 |- ( A. x e. { A , B , C } ph <-> A. x e. ( { A , B } u. { C } ) ph )
10		ralunb 3153	. . 3 |- ( A. x e. ( { A , B } u. { C } ) ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
11	9, 10	bitri 182	. 2 |- ( A. x e. { A , B , C } ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
12		df-3an 921	. 2 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
13	7, 11, 12	3bitr4g 221	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   u. cun 2971   {csn 3398   {cpr 3399   {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-tp 3406

This theorem is referenced by:  raltp  3449