Theorem ralcom3

Description: A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004) (Proof shortened by Wolf Lammen, 22-Dec-2024)

	Ref	Expression
Assertion	ralcom3	$$⊢\forall x\in A\left(x\in B\to \phi \right)\leftrightarrow \forall x\in B\left(x\in A\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$$⊢\left(x\in A\to \left(x\in B\to \phi \right)\right)\leftrightarrow \left(x\in B\to \left(x\in A\to \phi \right)\right)$$
2	1	ralbii2	$$⊢\forall x\in A\left(x\in B\to \phi \right)\leftrightarrow \forall x\in B\left(x\in A\to \phi \right)$$