Theorem raltp 3782

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 3778	. 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 1277	1 |- (A.x e. {A, B, C}ph <-> (ps /\ ch /\ th))

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934   = wceq 1642   e. wcel 1710  A.wral 2615  _Vcvv 2860  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by: (None)