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	⊢ (∀x ∈ A (x ∈ B → φ) ↔ ∀x ∈ B (x ∈ A → φ))

Proof of Theorem ralcom3

Step	Hyp	Ref	Expression
1		pm2.04 76	. . 3 ⊢ ((x ∈ A → (x ∈ B → φ)) → (x ∈ B → (x ∈ A → φ)))
2	1	ralimi2 2687	. 2 ⊢ (∀x ∈ A (x ∈ B → φ) → ∀x ∈ B (x ∈ A → φ))
3		pm2.04 76	. . 3 ⊢ ((x ∈ B → (x ∈ A → φ)) → (x ∈ A → (x ∈ B → φ)))
4	3	ralimi2 2687	. 2 ⊢ (∀x ∈ B (x ∈ A → φ) → ∀x ∈ A (x ∈ B → φ))
5	2, 4	impbii 180	1 ⊢ (∀x ∈ A (x ∈ B → φ) ↔ ∀x ∈ B (x ∈ A → φ))

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀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)