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

Theorem rexbid 2726
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
ralbid.1  |-  F/ x ph
ralbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rexbid  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)

Proof of Theorem rexbid
StepHypRef Expression
1 ralbid.1 . 2  |-  F/ x ph
2 ralbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
32adantr 453 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
41, 3rexbida 2722 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554    e. wcel 1726   E.wrex 2708
This theorem is referenced by:  rexbidv  2728  scott0  7815  infcvgaux1i  12641  rexeqbid  23969  bnj1463  29497
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 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-rex 2713
  Copyright terms: Public domain W3C validator