Theorem ralcom3 2477

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 2381	. 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 2381	. 2 |- ( A. x e. B ( x e. A -> ph ) -> A. x e. A ( x e. B -> ph ) )
5	2, 4	impbii 117	1 |- ( A. x e. A ( x e. B -> ph ) <-> A. x e. B ( x e. A -> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  zfregfr  4298