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Theorem rexpr 3582
Description: Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1  |-  A  e. 
_V
ralpr.2  |-  B  e. 
_V
ralpr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ralpr.4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
rexpr  |-  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) )
Distinct variable groups:    x, A    x, B    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem rexpr
StepHypRef Expression
1 ralpr.1 . 2  |-  A  e. 
_V
2 ralpr.2 . 2  |-  B  e. 
_V
3 ralpr.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 ralpr.4 . . 3  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
53, 4rexprg 3578 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
61, 2, 5mp2an 422 1  |-  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   E.wrex 2417   _Vcvv 2686   {cpr 3528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by: (None)
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