Theorem ralcom3 1315

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

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 31	. . 3 ⊢ ((x ∈ A → (x ∈ B → φ)) → (x ∈ B → (x ∈ A → φ)))
2	1	r19.20i2 1252	. 2 ⊢ (∀x ∈ A (x ∈ B → φ) → ∀x ∈ B (x ∈ A → φ))
3		pm2.04 31	. . 3 ⊢ ((x ∈ B → (x ∈ A → φ)) → (x ∈ A → (x ∈ B → φ)))
4	3	r19.20i2 1252	. 2 ⊢ (∀x ∈ B (x ∈ A → φ) → ∀x ∈ A (x ∈ B → φ))
5	2, 4	impbi 139	1 ⊢ (∀x ∈ A (x ∈ B → φ) ↔ ∀x ∈ B (x ∈ A → φ))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  ∀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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