Theorem rextpg 4237

Description: Convert a quantification over a triple to a disjunction. (Contributed 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
rextpg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. 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 rextpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
2		ralprg.2	. . . . . 6 |- ( x = B -> ( ph <-> ch ) )
3	1, 2	rexprg 4235	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
4	3	orbi1d 739	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ E. x e. { C } ph ) ) )
5		raltpg.3	. . . . . 6 |- ( x = C -> ( ph <-> th ) )
6	5	rexsng 4219	. . . . 5 |- ( C e. X -> ( E. x e. { C } ph <-> th ) )
7	6	orbi2d 738	. . . 4 |- ( C e. X -> ( ( ( ps \/ ch ) \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
8	4, 7	sylan9bb 736	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
9	8	3impa 1259	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
10		df-tp 4182	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
11	10	rexeqi 3143	. . 3 |- ( E. x e. { A , B , C } ph <-> E. x e. ( { A , B } u. { C } ) ph )
12		rexun 3793	. . 3 |- ( E. x e. ( { A , B } u. { C } ) ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
13	11, 12	bitri 264	. 2 |- ( E. x e. { A , B , C } ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
14		df-3or 1038	. 2 |- ( ( ps \/ ch \/ th ) <-> ( ( ps \/ ch ) \/ th ) )
15	9, 13, 14	3bitr4g 303	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  rextp  4241  fr3nr  6979  nb3grprlem2  26283  frgr3vlem2  27138  3vfriswmgr  27142