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Theorem rextp 3765
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 3761 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( ps  \/  ch  \/  th ) ) )
81, 2, 3, 7mp3an 1277 1  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( ps  \/  ch  \/  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ w3o 933    = wceq 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   {ctp 3718
This theorem is referenced by:  1cubr  20249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-v 2866  df-sbc 3068  df-un 3233  df-sn 3722  df-pr 3723  df-tp 3724
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