Theorem raltpg 4083

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 4081	. . . 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 4067	. . . 4 |- ( C e. X -> ( A. x e. { C } ph <-> th ) )
6	3, 5	bi2anan9 873	. . 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 1191	. 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 4037	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
9	8	raleqi 3058	. . 3 |- ( A. x e. { A , B , C } ph <-> A. x e. ( { A , B } u. { C } ) ph )
10		ralunb 3681	. . 3 |- ( A. x e. ( { A , B } u. { C } ) ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
11	9, 10	bitri 249	. 2 |- ( A. x e. { A , B , C } ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
12		df-3an 975	. 2 |- ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
13	7, 11, 12	3bitr4g 288	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   u. cun 3469   {csn 4032   {cpr 4034   {ctp 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-un 3476  df-sn 4033  df-pr 4035  df-tp 4037

This theorem is referenced by:  raltp  4087  raltpd  4155  f13dfv  6181  nb3grapr  24580  cusgra3v  24591  3v3e3cycl1  24771  constr3trllem2  24778  constr3trllem5  24781  frgra3v  25129