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Theorem eean 1853
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 1816 . . 3  |-  ( E. y ( ph  /\  ps )  <->  ( ph  /\  E. y ps ) )
32exbii 1569 . 2  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
4 eean.2 . . . 4  |-  F/ x ps
54nfex 1767 . . 3  |-  F/ x E. y ps
6519.41 1815 . 2  |-  ( E. x ( ph  /\  E. y ps )  <->  ( E. x ph  /\  E. y ps ) )
73, 6bitri 240 1  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531
This theorem is referenced by:  eeanv  1854  reean  2706  fnchoice  27700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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