Theorem rextpg 3647

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 3645	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
4	3	orbi1d 791	. . . 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 3634	. . . . 5 |- ( C e. X -> ( E. x e. { C } ph <-> th ) )
7	6	orbi2d 790	. . . 4 |- ( C e. X -> ( ( ( ps \/ ch ) \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
8	4, 7	sylan9bb 462	. . 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 1194	. 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 3601	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
11	10	rexeqi 2678	. . 3 |- ( E. x e. { A , B , C } ph <-> E. x e. ( { A , B } u. { C } ) ph )
12		rexun 3316	. . 3 |- ( E. x e. ( { A , B } u. { C } ) ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
13	11, 12	bitri 184	. 2 |- ( E. x e. { A , B , C } ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
14		df-3or 979	. 2 |- ( ( ps \/ ch \/ th ) <-> ( ( ps \/ ch ) \/ th ) )
15	9, 13, 14	3bitr4g 223	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   \/ w3o 977   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   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-3or 979  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-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-tp 3601

This theorem is referenced by:  rextp  3651