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Theorem aaan 1837
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 1818 . . 3  |-  ( A. y ( ph  /\  ps )  <->  ( ph  /\  A. y ps ) )
32albii 1556 . 2  |-  ( A. x A. y ( ph  /\ 
ps )  <->  A. x
( ph  /\  A. y ps ) )
4 aaan.2 . . . 4  |-  F/ x ps
54nfal 1778 . . 3  |-  F/ x A. y ps
6519.27 1817 . 2  |-  ( A. x ( ph  /\  A. y ps )  <->  ( A. x ph  /\  A. y ps ) )
73, 6bitri 240 1  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534
This theorem is referenced by:  mo  2178  2mo  2234  2eu4  2239  aaanv  27690  pm11.71  27699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1535
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