ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexeqdv Unicode version

Theorem rexeqdv 2608
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
Hypothesis
Ref Expression
raleq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
rexeqdv  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rexeqdv
StepHypRef Expression
1 raleq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 rexeq 2602 . 2  |-  ( A  =  B  ->  ( E. x  e.  A  ps 
<->  E. x  e.  B  ps ) )
31, 2syl 14 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   E.wrex 2392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397
This theorem is referenced by:  rexeqbidv  2614  rexeqbidva  2616  fnunirn  5634  cbvexfo  5653  fival  6824  genipv  7281  exfzdc  9957  infssuzex  11538  ennnfonelemrnh  11824  cnpfval  12259
  Copyright terms: Public domain W3C validator