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Theorem rmobida 3006
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 641 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3mobid 2282 . 2  |-  ( ph  ->  ( E* x ( x  e.  A  /\  ps )  <->  E* x ( x  e.  A  /\  ch ) ) )
5 df-rmo 2803 . 2  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
6 df-rmo 2803 . 2  |-  ( E* x  e.  A  ch  <->  E* x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 288 1  |-  ( ph  ->  ( E* x  e.  A  ps  <->  E* x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   F/wnf 1590    e. wcel 1758   E*wmo 2261   E*wrmo 2798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-eu 2264  df-mo 2265  df-rmo 2803
This theorem is referenced by:  rmobidva  3007  reuan  30144
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