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Theorem rextp 3866
Description: Convert a quantification over a triple to a disjunction. (Contributed 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
rextp  |-  ( E. 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 rextp
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, 6rextpg 3862 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( ps  \/  ch  \/  th ) ) )
81, 2, 3, 7mp3an 1280 1  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( ps  \/  ch  \/  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ w3o 936    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   {ctp 3818
This theorem is referenced by:  1cubr  20687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-tp 3824
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