Theorem cbvralfw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf with a disjoint variable condition, which does not require ax-10 , ax-13 . For a version not dependent on ax-11 and ax-12, see cbvralvw . (Contributed by NM, 7-Mar-2004) Avoid ax-10 , ax-13 . (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvralfw.1	$$⊢\underset{\_}{Ⅎ}xA$$
	cbvralfw.2	$$⊢\underset{\_}{Ⅎ}yA$$
	cbvralfw.3	$$⊢Ⅎy\phi$$
	cbvralfw.4	$$⊢Ⅎx\psi$$
	cbvralfw.5	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvralfw	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralfw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		cbvralfw.2	$$⊢\underset{\_}{Ⅎ}yA$$
3		cbvralfw.3	$$⊢Ⅎy\phi$$
4		cbvralfw.4	$$⊢Ⅎx\psi$$
5		cbvralfw.5	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
6	2	nfcri	$$⊢Ⅎyx\in A$$
7	6 3	nfim	$$⊢Ⅎy\left(x\in A\to \phi \right)$$
8	1	nfcri	$$⊢Ⅎxy\in A$$
9	8 4	nfim	$$⊢Ⅎx\left(y\in A\to \psi \right)$$
10		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
11	10 5	imbi12d	$$⊢x=y\to \left(\left(x\in A\to \phi \right)\leftrightarrow \left(y\in A\to \psi \right)\right)$$
12	7 9 11	cbvalv1	$$⊢\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall y\left(y\in A\to \psi \right)$$
13		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
14		df-ral	$$⊢\forall y\in A\psi \leftrightarrow \forall y\left(y\in A\to \psi \right)$$
15	12 13 14	3bitr4i	$$⊢\forall x\in A\phi \leftrightarrow \forall y\in A\psi$$