Theorem rextpg 3676

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 3674	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
4	3	orbi1d 792	. . . 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 3663	. . . . 5 |- ( C e. X -> ( E. x e. { C } ph <-> th ) )
7	6	orbi2d 791	. . . 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 1196	. 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 3630	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
11	10	rexeqi 2698	. . 3 |- ( E. x e. { A , B , C } ph <-> E. x e. ( { A , B } u. { C } ) ph )
12		rexun 3343	. . 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 981	. 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 709   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2167   E.wrex 2476   u. cun 3155   {csn 3622   {cpr 3623   {ctp 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-tp 3630

This theorem is referenced by:  rextp  3680