Theorem raltpg 3571

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 3569	. . . 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 3559	. . . 4 |- ( C e. X -> ( A. x e. { C } ph <-> th ) )
6	3, 5	bi2anan9 595	. . 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 1176	. 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 3530	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
9	8	raleqi 2628	. . 3 |- ( A. x e. { A , B , C } ph <-> A. x e. ( { A , B } u. { C } ) ph )
10		ralunb 3252	. . 3 |- ( A. x e. ( { A , B } u. { C } ) ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
11	9, 10	bitri 183	. 2 |- ( A. x e. { A , B , C } ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
12		df-3an 964	. 2 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
13	7, 11, 12	3bitr4g 222	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2414   u. cun 3064   {csn 3522   {cpr 3523   {ctp 3524

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-tp 3530

This theorem is referenced by:  raltp  3575  sumtp  11176