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Theorem aaanh 1494
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
Hypotheses
Ref Expression
aaanh.1  |-  ( ph  ->  A. y ph )
aaanh.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
aaanh  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )

Proof of Theorem aaanh
StepHypRef Expression
1 aaanh.1 . . . 4  |-  ( ph  ->  A. y ph )
2119.28h 1470 . . 3  |-  ( A. y ( ph  /\  ps )  <->  ( ph  /\  A. y ps ) )
32albii 1375 . 2  |-  ( A. x A. y ( ph  /\ 
ps )  <->  A. x
( ph  /\  A. y ps ) )
4 aaanh.2 . . . 4  |-  ( ps 
->  A. x ps )
54hbal 1382 . . 3  |-  ( A. y ps  ->  A. x A. y ps )
6519.27h 1468 . 2  |-  ( A. x ( ph  /\  A. y ps )  <->  ( A. x ph  /\  A. y ps ) )
73, 6bitri 177 1  |-  ( A. x A. y ( ph  /\ 
ps )  <->  ( A. x ph  /\  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  mo23  1957  2eu4  2009
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