Theorem raltp 3640

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 3636	. 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 1332	1 |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   {ctp 3585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  fztpval  10039