Theorem ralcom3 1315

Description: A commutative law for restricted quantifiers that swaps the domain of the restriction.

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 31	. . 3 |- ((x e. A -> (x e. B -> ph)) -> (x e. B -> (x e. A -> ph)))
2	1	r19.20i2 1252	. 2 |- (A.x e. A (x e. B -> ph) -> A.x e. B (x e. A -> ph))
3		pm2.04 31	. . 3 |- ((x e. B -> (x e. A -> ph)) -> (x e. A -> (x e. B -> ph)))
4	3	r19.20i2 1252	. 2 |- (A.x e. B (x e. A -> ph) -> A.x e. A (x e. B -> ph))
5	2, 4	impbi 139	1 |- (A.x e. A (x e. B -> ph) <-> A.x e. B (x e. A -> ph))

Syntax hints:   -> wi 2   <-> wb 127   e. wcel 1092  A.wral 1201

This theorem is referenced by:  find 2396

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-gen 677

This theorem depends on definitions:  df-bi 128  df-ral 1205

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