Theorem raltp 3700

Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised 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
raltp	|- ( A. 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 raltp

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	raltpg 3696	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )
8	1, 2, 3, 7	mp3an 1350	1 |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   {ctp 3645

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-sbc 3006  df-un 3178  df-sn 3649  df-pr 3650  df-tp 3651

This theorem is referenced by:  fztpval  10240