MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexpr Structured version   Unicode version

Theorem rexpr 3949
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 3945 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
61, 2, 5mp2an 672 1  |-  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1369    e. wcel 1756   E.wrex 2735   _Vcvv 2991   {cpr 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2740  df-v 2993  df-sbc 3206  df-un 3352  df-sn 3897  df-pr 3899
This theorem is referenced by:  xpsdsval  19975
  Copyright terms: Public domain W3C validator