Theorem cbvralvw

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. Version of cbvralv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 28-Jan-1997) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvralvw	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	imbi12d	$$⊢x=y\to \left(\left(x\in A\to \phi \right)\leftrightarrow \left(y\in A\to \psi \right)\right)$$
4	3	cbvalvw	$$⊢\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall y\left(y\in A\to \psi \right)$$
5		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
6		df-ral	$$⊢\forall y\in A\psi \leftrightarrow \forall y\left(y\in A\to \psi \right)$$
7	4 5 6	3bitr4i	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\psi$$