Theorem cbvrexcsf

Description: A more general version of cbvrexf that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Andrew Salmon, 13-Jul-2011) (Proof shortened by Mario Carneiro, 7-Dec-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvralcsf.1	$$⊢\underset{\_}{Ⅎ}yA$$
	cbvralcsf.2	$$⊢\underset{\_}{Ⅎ}xB$$
	cbvralcsf.3	$$⊢Ⅎy\phi$$
	cbvralcsf.4	$$⊢Ⅎx\psi$$
	cbvralcsf.5	$$⊢x=y\to A=B$$
	cbvralcsf.6	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrexcsf	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralcsf.1	$$⊢\underset{\_}{Ⅎ}yA$$
2		cbvralcsf.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		cbvralcsf.3	$$⊢Ⅎy\phi$$
4		cbvralcsf.4	$$⊢Ⅎx\psi$$
5		cbvralcsf.5	$$⊢x=y\to A=B$$
6		cbvralcsf.6	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
7	3	nfn	$$⊢Ⅎy\neg \phi$$
8	4	nfn	$$⊢Ⅎx\neg \psi$$
9	6	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
10	1 2 7 8 5 9	cbvralcsf	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall y\in B\neg \psi$$
11	10	notbii	$$⊢\neg \forall x\in A\neg \phi \leftrightarrow \neg \forall y\in B\neg \psi$$
12		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
13		dfrex2	$$⊢\exists y\in B\psi \leftrightarrow \neg \forall y\in B\neg \psi$$
14	11 12 13	3bitr4i	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in B\psi$$