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Theorem rexbii2 2352
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 1512 . 2  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ps )
)
3 df-rex 2329 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-rex 2329 . 2  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
52, 3, 43bitr4i 205 1  |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by:  rexeqbii  2354  rexbiia  2356  rexrab  2727  rexdifsn  3527  bnd2  3954  rexuz2  8620  rexrp  8703  rexuz3  9817
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