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Theorem rmobida 2897
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 624 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3mobid 2317 . 2  |-  ( ph  ->  ( E* x ( x  e.  A  /\  ps )  <->  E* x ( x  e.  A  /\  ch ) ) )
5 df-rmo 2715 . 2  |-  ( E* x  e.  A ps  <->  E* x ( x  e.  A  /\  ps )
)
6 df-rmo 2715 . 2  |-  ( E* x  e.  A ch  <->  E* x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 281 1  |-  ( ph  ->  ( E* x  e.  A ps  <->  E* x  e.  A ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   F/wnf 1554    e. wcel 1726   E*wmo 2284   E*wrmo 2710
This theorem is referenced by:  rmobidva  2898  reuan  27936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-eu 2287  df-mo 2288  df-rmo 2715
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