Theorem ralcom3 3105

Description: A commutation law for restricted universal quantifiers that swaps the domains 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 90	. . 3 |- ( ( x e. A -> ( x e. B -> ph ) ) -> ( x e. B -> ( x e. A -> ph ) ) )
2	1	ralimi2 2949	. 2 |- ( A. x e. A ( x e. B -> ph ) -> A. x e. B ( x e. A -> ph ) )
3		pm2.04 90	. . 3 |- ( ( x e. B -> ( x e. A -> ph ) ) -> ( x e. A -> ( x e. B -> ph ) ) )
4	3	ralimi2 2949	. 2 |- ( A. x e. B ( x e. A -> ph ) -> A. x e. A ( x e. B -> ph ) )
5	2, 4	impbii 199	1 |- ( A. x e. A ( x e. B -> ph ) <-> A. x e. B ( x e. A -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ral 2917

This theorem is referenced by:  tgss2  20791  ist1-3  21153  isreg2  21181