Theorem raltp 3649

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 3645	. 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 1337	1 |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   {ctp 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-tp 3600

This theorem is referenced by:  fztpval  10078