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Theorem rexbii2 2335
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 1496 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ps )
)
3 df-rex 2312 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-rex 2312 . 2  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
52, 3, 43bitr4i 201 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98   E.wex 1381    e. wcel 1393   E.wrex 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-rex 2312
This theorem is referenced by:  rexeqbii  2337  rexbiia  2339  rexrab  2704  rexdifsn  3499  bnd2  3926  rexuz2  8524  rexrp  8605  rexuz3  9588
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