Theorem cbvex2

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 cbvex2v if possible. (Contributed by NM, 14-Sep-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 16-Jun-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval2.1	$$⊢Ⅎz\phi$$
	cbval2.2	$$⊢Ⅎw\phi$$
	cbval2.3	$$⊢Ⅎx\psi$$
	cbval2.4	$$⊢Ⅎy\psi$$
	cbval2.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvex2	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval2.1	$$⊢Ⅎz\phi$$
2		cbval2.2	$$⊢Ⅎw\phi$$
3		cbval2.3	$$⊢Ⅎx\psi$$
4		cbval2.4	$$⊢Ⅎy\psi$$
5		cbval2.5	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
6	1	nfn	$$⊢Ⅎz\neg \phi$$
7	2	nfn	$$⊢Ⅎw\neg \phi$$
8	3	nfn	$$⊢Ⅎx\neg \psi$$
9	4	nfn	$$⊢Ⅎy\neg \psi$$
10	5	notbid	$$⊢\left(x=z\wedge y=w\right)\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
11	6 7 8 9 10	cbval2	$$⊢\forall x\forall y\neg \phi \leftrightarrow \forall z\forall w\neg \psi$$
12		2nexaln	$$⊢\neg \exists x\exists y\phi \leftrightarrow \forall x\forall y\neg \phi$$
13		2nexaln	$$⊢\neg \exists z\exists w\psi \leftrightarrow \forall z\forall w\neg \psi$$
14	11 12 13	3bitr4i	$$⊢\neg \exists x\exists y\phi \leftrightarrow \neg \exists z\exists w\psi$$
15	14	con4bii	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$