Theorem rextpg 3759

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 3757	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
4	3	orbi1d 803	. . . 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 3746	. . . . 5 |- ( C e. X -> ( E. x e. { C } ph <-> th ) )
7	6	orbi2d 802	. . . 4 |- ( C e. X -> ( ( ( ps \/ ch ) \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
8	4, 7	sylan9bb 466	. . 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 1225	. 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 3713	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
11	10	rexeqi 2754	. . 3 |- ( E. x e. { A , B , C } ph <-> E. x e. ( { A , B } u. { C } ) ph )
12		rexun 3409	. . 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 1010	. 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 720   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   u. cun 3218   {csn 3705   {cpr 3706   {ctp 3707

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-tp 3713

This theorem is referenced by:  rextp  3763