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Theorem rexbi 2491
Description: Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
Assertion
Ref Expression
rexbi  |-  ( A. x  e.  A  ( ph 
<->  ps )  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  ps ) )

Proof of Theorem rexbi
StepHypRef Expression
1 nfra1 2398 . 2  |-  F/ x A. x  e.  A  ( ph  <->  ps )
2 rsp 2412 . . 3  |-  ( A. x  e.  A  ( ph 
<->  ps )  ->  (
x  e.  A  -> 
( ph  <->  ps ) ) )
32imp 122 . 2  |-  ( ( A. x  e.  A  ( ph  <->  ps )  /\  x  e.  A )  ->  ( ph 
<->  ps ) )
41, 3rexbida 2364 1  |-  ( A. x  e.  A  ( ph 
<->  ps )  ->  ( E. x  e.  A  ph  <->  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434   A.wral 2349   E.wrex 2350
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  rexrnmpt2  5647
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