Theorem raleq

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rexeq	$$⊢A=B\to \left(\exists x\in A\neg \phi \leftrightarrow \exists x\in B\neg \phi \right)$$
2		rexnal	$$⊢\exists x\in A\neg \phi \leftrightarrow \neg \forall x\in A\phi$$
3		rexnal	$$⊢\exists x\in B\neg \phi \leftrightarrow \neg \forall x\in B\phi$$
4	1 2 3	3bitr3g	$$⊢A=B\to \left(\neg \forall x\in A\phi \leftrightarrow \neg \forall x\in B\phi \right)$$
5	4	con4bid	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\phi \right)$$