Theorem aaan 2170

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 2096	. . 3 |- ( A. y ( ph /\ ps ) <-> ( ph /\ A. y ps ) )
3	2	albii 1747	. 2 |- ( A. x A. y ( ph /\ ps ) <-> A. x ( ph /\ A. y ps ) )
4		aaan.2	. . . 4 |- F/ x ps
5	4	nfal 2153	. . 3 |- F/ x A. y ps
6	5	19.27 2095	. 2 |- ( A. x ( ph /\ A. y ps ) <-> ( A. x ph /\ A. y ps ) )
7	3, 6	bitri 264	1 |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-mo3OLD  32832  aaanv  38588  pm11.71  38597