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Theorem rexbi 2542
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 2443 . 2  |-  F/ x A. x  e.  A  ( ph  <->  ps )
2 rsp 2457 . . 3  |-  ( A. x  e.  A  ( ph 
<->  ps )  ->  (
x  e.  A  -> 
( ph  <->  ps ) ) )
32imp 123 . 2  |-  ( ( A. x  e.  A  ( ph  <->  ps )  /\  x  e.  A )  ->  ( ph 
<->  ps ) )
41, 3rexbida 2409 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 104    e. wcel 1465   A.wral 2393   E.wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-ral 2398  df-rex 2399
This theorem is referenced by:  rexrnmpo  5854
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