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Theorem ceqsrexv2 28604
Description: Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
Hypothesis
Ref Expression
ceqsrexv2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsrexv2  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsrexv2
StepHypRef Expression
1 ceqsrexv2.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21ceqsrexbv 3238 1  |-  ( E. x  e.  B  ( x  =  A  /\  ph )  <->  ( A  e.  B  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2820  df-v 3115
This theorem is referenced by: (None)
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