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Theorem rmobiia 2730
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 618 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32mobii 2179 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  A  /\  ps )
)
4 df-rmo 2551 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
5 df-rmo 2551 . 2  |-  ( E* x  e.  A ps  <->  E* x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 268 1  |-  ( E* x  e.  A ph  <->  E* x  e.  A ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E*wmo 2144   E*wrmo 2546
This theorem is referenced by:  rmobii  2731
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-tru 1310  df-ex 1529  df-nf 1532  df-eu 2147  df-mo 2148  df-rmo 2551
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