Theorem cbvrexf

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexfw when possible. (Contributed by FL, 27-Apr-2008) (Revised by Mario Carneiro, 9-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvralf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		cbvralf.2	$$⊢\underset{\_}{Ⅎ}yA$$
3		cbvralf.3	$$⊢Ⅎy\phi$$
4		cbvralf.4	$$⊢Ⅎx\psi$$
5		cbvralf.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	cbvralf	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall y\in A\neg \psi$$
10	9	notbii	$$⊢\neg \forall x\in A\neg \phi \leftrightarrow \neg \forall y\in A\neg \psi$$
11		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
12		dfrex2	$$⊢\exists y\in A\psi \leftrightarrow \neg \forall y\in A\neg \psi$$
13	10 11 12	3bitr4i	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$