Theorem cbvex4v

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 cbvex4vw if possible. (Contributed by NM, 26-Jul-1995) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvex4v.1	$$⊢\left(x=v\wedge y=u\right)\to \left(\phi \leftrightarrow \psi \right)$$
	cbvex4v.2	$$⊢\left(z=f\wedge w=g\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvex4v	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists f\exists g\chi$$

Proof

Step	Hyp	Ref	Expression
1		cbvex4v.1	$$⊢\left(x=v\wedge y=u\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		cbvex4v.2	$$⊢\left(z=f\wedge w=g\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	1	2exbidv	$$⊢\left(x=v\wedge y=u\right)\to \left(\exists z\exists w\phi \leftrightarrow \exists z\exists w\psi \right)$$
4	3	cbvex2vv	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists z\exists w\psi$$
5	2	cbvex2vv	$$⊢\exists z\exists w\psi \leftrightarrow \exists f\exists g\chi$$
6	5	2exbii	$$⊢\exists v\exists u\exists z\exists w\psi \leftrightarrow \exists v\exists u\exists f\exists g\chi$$
7	4 6	bitri	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists f\exists g\chi$$