Theorem ralcom3 1777

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 30	. . 3 |- ((x e. A -> (x e. B -> ph)) -> (x e. B -> (x e. A -> ph)))
2	1	r19.20i2 1703	. 2 |- (A.x e. A (x e. B -> ph) -> A.x e. B (x e. A -> ph))
3		pm2.04 30	. . 3 |- ((x e. B -> (x e. A -> ph)) -> (x e. A -> (x e. B -> ph)))
4	3	r19.20i2 1703	. 2 |- (A.x e. B (x e. A -> ph) -> A.x e. A (x e. B -> ph))
5	2, 4	impbi 157	1 |- (A.x e. A (x e. B -> ph) <-> A.x e. B (x e. A -> ph))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  A.wral 1645

This theorem is referenced by:  find 3155

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-ral 1649

Copyright terms: Public domain