Theorem aaan 1884

Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Hypotheses

Ref	Expression
aaan.1	|- F/yph
aaan.2	|- F/xps

Assertion

Ref	Expression
aaan	|- (A.xA.y(ph /\ ps) <-> (A.xph /\ A.yps))

Proof of Theorem aaan

Step	Hyp	Ref	Expression
1		aaan.1	. . . 4 |- F/yph
2	1	19.28 1870	. . 3 |- (A.y(ph /\ ps) <-> (ph /\ A.yps))
3	2	albii 1566	. 2 |- (A.xA.y(ph /\ ps) <-> A.x(ph /\ A.yps))
4		aaan.2	. . . 4 |- F/xps
5	4	nfal 1842	. . 3 |- F/xA.yps
6	5	19.27 1869	. 2 |- (A.x(ph /\ A.yps) <-> (A.xph /\ A.yps))
7	3, 6	bitri 240	1 |- (A.xA.y(ph /\ ps) <-> (A.xph /\ A.yps))

Syntax hints:   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  mo  2226  2mo  2282  2eu4  2287