Theorem ralcom3 2777

Description: A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
ralcom3	|- (A.x e. A (x e. B -> ph) <-> A.x e. B (x e. A -> ph))

Proof of Theorem ralcom3

Step	Hyp	Ref	Expression
1		pm2.04 76	. . 3 |- ((x e. A -> (x e. B -> ph)) -> (x e. B -> (x e. A -> ph)))
2	1	ralimi2 2687	. 2 |- (A.x e. A (x e. B -> ph) -> A.x e. B (x e. A -> ph))
3		pm2.04 76	. . 3 |- ((x e. B -> (x e. A -> ph)) -> (x e. A -> (x e. B -> ph)))
4	3	ralimi2 2687	. 2 |- (A.x e. B (x e. A -> ph) -> A.x e. A (x e. B -> ph))
5	2, 4	impbii 180	1 |- (A.x e. A (x e. B -> ph) <-> A.x e. B (x e. A -> ph))

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by: (None)