Theorem rextp 3724

Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
raltp.1	|- A e. _V
raltp.2	|- B e. _V
raltp.3	|- C e. _V
raltp.4	|- ( x = A -> ( ph <-> ps ) )
raltp.5	|- ( x = B -> ( ph <-> ch ) )
raltp.6	|- ( x = C -> ( ph <-> th ) )

Assertion

Ref	Expression
rextp	|- ( 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 hint:   ph( x)

Proof of Theorem rextp

Step	Hyp	Ref	Expression
1		raltp.1	. 2 |- A e. _V
2		raltp.2	. 2 |- B e. _V
3		raltp.3	. 2 |- C e. _V
4		raltp.4	. . 3 |- ( x = A -> ( ph <-> ps ) )
5		raltp.5	. . 3 |- ( x = B -> ( ph <-> ch ) )
6		raltp.6	. . 3 |- ( x = C -> ( ph <-> th ) )
7	4, 5, 6	rextpg 3720	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )
8	1, 2, 3, 7	mp3an 1371	1 |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ w3o 1001   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   {ctp 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-tp 3674

This theorem is referenced by: (None)