MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aaan Unicode version

Theorem aaan 1825
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 1806 . . 3  |-  ( A. y ( ph  /\  ps )  <->  ( ph  /\  A. y ps ) )
32albii 1553 . 2  |-  ( A. x A. y ( ph  /\ 
ps )  <->  A. x
( ph  /\  A. y ps ) )
4 aaan.2 . . . 4  |-  F/ x ps
54nfal 1766 . . 3  |-  F/ x A. y ps
6519.27 1805 . 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 1527   F/wnf 1531
This theorem is referenced by:  mo  2165  2mo  2221  2eu4  2226  aaanv  27587  pm11.71  27596
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-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1532
  Copyright terms: Public domain W3C validator