Theorem aaan 794

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	19.28 751	. . 3 |- (A.y(ph /\ ps) <-> (ph /\ A.yps))
3	2	bial 695	. 2 |- (A.xA.y(ph /\ ps) <-> A.x(ph /\ A.yps))
4		aaan.2	. . . 4 |- (ps -> A.xps)
5	4	hbal 700	. . 3 |- (A.yps -> A.xA.yps)
6	5	19.27 750	. 2 |- (A.x(ph /\ A.yps) <-> (A.xph /\ A.yps))
7	3, 6	bitr 151	1 |- (A.xA.y(ph /\ ps) <-> (A.xph /\ A.yps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672

This theorem is referenced by:  mo 1020  2eu4 1070

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198

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