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Theorem rmobiia 2516
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmobiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rmobiia  |-  ( E* x  e.  A  ph  <->  E* x  e.  A  ps )

Proof of Theorem rmobiia
StepHypRef Expression
1 rmobiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 435 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32mobii 1953 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  A  /\  ps )
)
4 df-rmo 2331 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
5 df-rmo 2331 . 2  |-  ( E* x  e.  A  ps  <->  E* x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 205 1  |-  ( E* x  e.  A  ph  <->  E* x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    e. wcel 1409   E*wmo 1917   E*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-eu 1919  df-mo 1920  df-rmo 2331
This theorem is referenced by:  rmobii  2517
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