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Theorem rexpr 3822
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 3818 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
61, 2, 5mp2an 654 1  |-  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916   {cpr 3775
This theorem is referenced by:  xpsdsval  18364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-sbc 3122  df-un 3285  df-sn 3780  df-pr 3781
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