Theorem cbvrexfw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf with a disjoint variable condition, which does not require ax-13 . For a version not dependent on ax-11 and ax-12, see cbvrexvw . (Contributed by FL, 27-Apr-2008) Avoid ax-10 , ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvrexfw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		cbvrexfw.2	$$⊢\underset{\_}{Ⅎ}yA$$
3		cbvrexfw.3	$$⊢Ⅎy\phi$$
4		cbvrexfw.4	$$⊢Ⅎx\psi$$
5		cbvrexfw.5	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
6	3	nfn	$$⊢Ⅎy\neg \phi$$
7	4	nfn	$$⊢Ⅎx\neg \psi$$
8	5	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
9	1 2 6 7 8	cbvralfw	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall y\in A\neg \psi$$
10		ralnex	$$⊢\forall x\in A\neg \phi \leftrightarrow \neg \exists x\in A\phi$$
11		ralnex	$$⊢\forall y\in A\neg \psi \leftrightarrow \neg \exists y\in A\psi$$
12	9 10 11	3bitr3i	$$⊢\neg \exists x\in A\phi \leftrightarrow \neg \exists y\in A\psi$$
13	12	con4bii	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$