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Theorem aaan 1520
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 1496 . . 3  |-  ( A. y ( ph  /\  ps )  <->  ( ph  /\  A. y ps ) )
32albii 1400 . 2  |-  ( A. x A. y ( ph  /\ 
ps )  <->  A. x
( ph  /\  A. y ps ) )
4 aaan.2 . . . 4  |-  F/ x ps
54nfal 1509 . . 3  |-  F/ x A. y ps
6519.27 1494 . 2  |-  ( A. x ( ph  /\  A. y ps )  <->  ( A. x ph  /\  A. y ps ) )
73, 6bitri 182 1  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1283   F/wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by: (None)
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