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Theorem aaan 1909
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
Hypotheses
Ref Expression
aaan.1  |-  F/ y
ph
aaan.2  |-  F/ x ps
Assertion
Ref Expression
aaan  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )

Proof of Theorem aaan
StepHypRef Expression
1 aaan.1 . . . 4  |-  F/ y
ph
2119.28 1844 . . 3  |-  ( A. y ( ph  /\  ps )  <->  ( ph  /\  A. y ps ) )
32albii 1576 . 2  |-  ( A. x A. y ( ph  /\ 
ps )  <->  A. x
( ph  /\  A. y ps ) )
4 aaan.2 . . . 4  |-  F/ x ps
54nfal 1866 . . 3  |-  F/ x A. y ps
6519.27 1843 . 2  |-  ( A. x ( ph  /\  A. y ps )  <->  ( A. x ph  /\  A. y ps ) )
73, 6bitri 242 1  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   A.wal 1550   F/wnf 1554
This theorem is referenced by:  mo  2309  2mo  2365  2eu4  2370  aaanv  27602  pm11.71  27611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-7 1751  ax-11 1763
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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