Theorem aaan 1580

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

Step	Hyp	Ref	Expression
1		aaan.1	. . . 4 |- F/ y ph
2	1	19.28 1556	. . 3 |- ( A. y ( ph /\ ps ) <-> ( ph /\ A. y ps ) )
3	2	albii 1463	. 2 |- ( A. x A. y ( ph /\ ps ) <-> A. x ( ph /\ A. y ps ) )
4		aaan.2	. . . 4 |- F/ x ps
5	4	nfal 1569	. . 3 |- F/ x A. y ps
6	5	19.27 1554	. 2 |- ( A. x ( ph /\ A. y ps ) <-> ( A. x ph /\ A. y ps ) )
7	3, 6	bitri 183	1 |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1346   F/wnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by: (None)