Theorem ralcom3 1824

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 30	. . 3 ⊢ ((x ∈ A → (x ∈ B → φ)) → (x ∈ B → (x ∈ A → φ)))
2	1	r19.20i2 1750	. 2 ⊢ (∀x ∈ A (x ∈ B → φ) → ∀x ∈ B (x ∈ A → φ))
3		pm2.04 30	. . 3 ⊢ ((x ∈ B → (x ∈ A → φ)) → (x ∈ A → (x ∈ B → φ)))
4	3	r19.20i2 1750	. 2 ⊢ (∀x ∈ B (x ∈ A → φ) → ∀x ∈ A (x ∈ B → φ))
5	2, 4	impbii 164	1 ⊢ (∀x ∈ A (x ∈ B → φ) ↔ ∀x ∈ B (x ∈ A → φ))

Syntax hints:   → wi 3   ↔ wb 153   ∈ wcel 999  ∀wral 1692

This theorem is referenced by:  find 3212

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-4 1014  ax-5o 1016

This theorem depends on definitions:  df-bi 154  df-ral 1696

Copyright terms: Public domain