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Theorem eean 1901
Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
eean.1  |-  F/ y
ph
eean.2  |-  F/ x ps
Assertion
Ref Expression
eean  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )

Proof of Theorem eean
StepHypRef Expression
1 eean.1 . . . 4  |-  F/ y
ph
2119.42 1666 . . 3  |-  ( E. y ( ph  /\  ps )  <->  ( ph  /\  E. y ps ) )
32exbii 1584 . 2  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
4 eean.2 . . . 4  |-  F/ x ps
54nfex 1616 . . 3  |-  F/ x E. y ps
6519.41 1664 . 2  |-  ( E. x ( ph  /\  E. y ps )  <->  ( E. x ph  /\  E. y ps ) )
73, 6bitri 183 1  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   F/wnf 1436   E.wex 1468
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  eeanv  1902  reean  2597
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