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Theorem rexbii2 2421
Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
rexbii2.1  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ps )
)
Assertion
Ref Expression
rexbii2  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ps )

Proof of Theorem rexbii2
StepHypRef Expression
1 rexbii2.1 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ps )
)
21exbii 1567 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ps )
)
3 df-rex 2397 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-rex 2397 . 2  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
52, 3, 43bitr4i 211 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1451    e. wcel 1463   E.wrex 2392
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-rex 2397
This theorem is referenced by:  rexeqbii  2423  rexbiia  2425  rexrab  2818  rexdifsn  3623  bnd2  4065  suplocsrlemb  7578  rexuz2  9325  rexrp  9412  rexuz3  10702
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