Theorem aaanh 1565

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

Step	Hyp	Ref	Expression
1		aaanh.1	. . . 4 |- ( ph -> A. y ph )
2	1	19.28h 1541	. . 3 |- ( A. y ( ph /\ ps ) <-> ( ph /\ A. y ps ) )
3	2	albii 1446	. 2 |- ( A. x A. y ( ph /\ ps ) <-> A. x ( ph /\ A. y ps ) )
4		aaanh.2	. . . 4 |- ( ps -> A. x ps )
5	4	hbal 1453	. . 3 |- ( A. y ps -> A. x A. y ps )
6	5	19.27h 1539	. 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:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329

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 1423  ax-7 1424  ax-gen 1425  ax-4 1487

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mo23  2040  2eu4  2092