Theorem aaan 1160

Description: Rearrange universal quantifiers.

Hypotheses

Ref	Expression
aaan.1	|- (ph -> A.yph)
aaan.2	|- (ps -> A.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 |- (ph -> A.yph)
2	1	19.28 1111	. . 3 |- (A.y(ph /\ ps) <-> (ph /\ A.yps))
3	2	albii 1040	. 2 |- (A.xA.y(ph /\ ps) <-> A.x(ph /\ A.yps))
4		aaan.2	. . . 4 |- (ps -> A.xps)
5	4	hbal 1046	. . 3 |- (A.yps -> A.xA.yps)
6	5	19.27 1110	. 2 |- (A.x(ph /\ A.yps) <-> (A.xph /\ A.yps))
7	3, 6	bitri 180	1 |- (A.xA.y(ph /\ ps) <-> (A.xph /\ A.yps))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995

This theorem is referenced by:  mo 1435  2mo 1490  2eu4 1495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-4 1014  ax-5o 1016

This theorem depends on definitions:  df-bi 154  df-an 232

Copyright terms: Public domain