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Theorem eean 2092
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 2071 . . 3  |-  ( E. y ( ph  /\  ps )  <->  ( ph  /\  E. y ps ) )
32exbii 1726 . 2  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
4 eean.2 . . . 4  |-  F/ x ps
54nfex 2050 . . 3  |-  F/ x E. y ps
6519.41 2070 . 2  |-  ( E. x ( ph  /\  E. y ps )  <->  ( E. x ph  /\  E. y ps ) )
73, 6bitri 257 1  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   E.wex 1671   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  eeanv  2093  reean  2943
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