Theorem raltpg 3646

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 3644	. . . 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 3633	. . . 4 |- ( C e. X -> ( A. x e. { C } ph <-> th ) )
6	3, 5	bi2anan9 606	. . 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 1194	. 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 3601	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
9	8	raleqi 2677	. . 3 |- ( A. x e. { A , B , C } ph <-> A. x e. ( { A , B } u. { C } ) ph )
10		ralunb 3317	. . 3 |- ( A. x e. ( { A , B } u. { C } ) ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
11	9, 10	bitri 184	. 2 |- ( A. x e. { A , B , C } ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
12		df-3an 980	. 2 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
13	7, 11, 12	3bitr4g 223	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   u. cun 3128   {csn 3593   {cpr 3594   {ctp 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 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-tp 3601

This theorem is referenced by:  raltp  3650  sumtp  11422