Theorem rextp 3731

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 3727	. 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 1374	1 |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ w3o 1004   = wceq 1398   e. wcel 2202   E.wrex 2512   _Vcvv 2803   {ctp 3675

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-sn 3679  df-pr 3680  df-tp 3681

This theorem is referenced by: (None)