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Theorem rexbi 2498
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 2405 . 2  |-  F/ x A. x  e.  A  ( ph  <->  ps )
2 rsp 2419 . . 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 2371 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 1436   A.wral 2355   E.wrex 2356
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-ral 2360  df-rex 2361
This theorem is referenced by:  rexrnmpt2  5717
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