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Theorem raltp 3855
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 3851 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) ) )
81, 2, 3, 7mp3an 1279 1  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   {ctp 3808
This theorem is referenced by:  fztpval  11097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-tp 3814
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