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Theorem raltp 3688
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
StepHypRef 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 ) )
74, 5, 6raltpg 3684 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) ) )
81, 2, 3, 7mp3an 1277 1  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   {ctp 3642
This theorem is referenced by:  fztpval  10845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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