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Theorem rmobida 2553
Description: Formula-building rule for restricted existential quantifier (deduction form). (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 440 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
41, 3mobid 1983 . 2  |-  ( ph  ->  ( E* x ( x  e.  A  /\  ps )  <->  E* x ( x  e.  A  /\  ch ) ) )
5 df-rmo 2367 . 2  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
6 df-rmo 2367 . 2  |-  ( E* x  e.  A  ch  <->  E* x ( x  e.  A  /\  ch )
)
74, 5, 63bitr4g 221 1  |-  ( ph  ->  ( E* x  e.  A  ps  <->  E* x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   F/wnf 1394    e. wcel 1438   E*wmo 1949   E*wrmo 2362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951  df-mo 1952  df-rmo 2367
This theorem is referenced by:  rmobidva  2554
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