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

Theorem rmobida 2727
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmobida.1  |-  F/ x ph
rmobida.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rmobida  |-  ( ph  ->  ( E* x  e.  A ps  <->  E* x  e.  A ch ) )

Proof of Theorem rmobida
StepHypRef Expression
1 rmobida.1 . . 3  |-  F/ x ph
2 rmobida.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32pm5.32da 622 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3mobid 2177 . 2  |-  ( ph  ->  ( E* x ( x  e.  A  /\  ps )  <->  E* x ( x  e.  A  /\  ch ) ) )
5 df-rmo 2551 . 2  |-  ( E* x  e.  A ps  <->  E* x ( x  e.  A  /\  ps )
)
6 df-rmo 2551 . 2  |-  ( E* x  e.  A ch  <->  E* x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 279 1  |-  ( ph  ->  ( E* x  e.  A ps  <->  E* x  e.  A ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    e. wcel 1684   E*wmo 2144   E*wrmo 2546
This theorem is referenced by:  rmobidva  2728  reuan  27958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-eu 2147  df-mo 2148  df-rmo 2551
  Copyright terms: Public domain W3C validator