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

Theorem rexsn 3874
Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ralsn.1  |-  A  e. 
_V
ralsn.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexsn  |-  ( E. x  e.  { A } ph  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rexsn
StepHypRef Expression
1 ralsn.1 . 2  |-  A  e. 
_V
2 ralsn.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32rexsng 3871 . 2  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 5 1  |-  ( E. x  e.  { A } ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727   E.wrex 2712   _Vcvv 2962   {csn 3838
This theorem is referenced by:  elsnres  5211  oarec  6834  snec  6996  zornn0g  8416  fpwwe2lem13  8548  elreal  9037  vdwlem6  13385  restsn  17265  snclseqg  18176  ust0  18280  eldm3  25416  heiborlem3  26560
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 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-v 2964  df-sbc 3168  df-sn 3844
  Copyright terms: Public domain W3C validator