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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
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 3645 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) ) )
81, 2, 3, 7mp3an 1337 1  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( ps  /\  ch  /\ 
th ) )
Colors of variables: wff set class
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
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