Theorem cbvex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvexvw , cbvexv1 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval.1	$$⊢Ⅎy\phi$$
	cbval.2	$$⊢Ⅎx\psi$$
	cbval.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvex	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval.1	$$⊢Ⅎy\phi$$
2		cbval.2	$$⊢Ⅎx\psi$$
3		cbval.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	nfn	$$⊢Ⅎy\neg \phi$$
5	2	nfn	$$⊢Ⅎx\neg \psi$$
6	3	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
7	4 5 6	cbval	$$⊢\forall x\neg \phi \leftrightarrow \forall y\neg \psi$$
8		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
9		alnex	$$⊢\forall y\neg \psi \leftrightarrow \neg \exists y\psi$$
10	7 8 9	3bitr3i	$$⊢\neg \exists x\phi \leftrightarrow \neg \exists y\psi$$
11	10	con4bii	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$