Theorem aaan 1902

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 1838	. . 3 |- ( A. y ( ph /\ ps ) <-> ( ph /\ A. y ps ) )
3	2	albii 1572	. 2 |- ( A. x A. y ( ph /\ ps ) <-> A. x ( ph /\ A. y ps ) )
4		aaan.2	. . . 4 |- F/ x ps
5	4	nfal 1860	. . 3 |- F/ x A. y ps
6	5	19.27 1837	. 2 |- ( A. x ( ph /\ A. y ps ) <-> ( A. x ph /\ A. y ps ) )
7	3, 6	bitri 241	1 |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) )

Syntax hints:   <-> wb 177   /\ wa 359   A.wal 1546   F/wnf 1550

This theorem is referenced by:  mo  2284  2mo  2340  2eu4  2345  aaanv  27463  pm11.71  27472

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-7 1745  ax-11 1757

This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551