Theorem cbvex4vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex4v with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvex4vw.1	$$⊢\left(x=v\wedge y=u\right)\to \left(\phi \leftrightarrow \psi \right)$$
	cbvex4vw.2	$$⊢\left(z=f\wedge w=g\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvex4vw	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists f\exists g\chi$$

Proof

Step	Hyp	Ref	Expression
1		cbvex4vw.1	$$⊢\left(x=v\wedge y=u\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		cbvex4vw.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	cbvex2vw	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists z\exists w\psi$$
5	2	cbvex2vw	$$⊢\exists z\exists w\psi \leftrightarrow \exists f\exists g\chi$$
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	$$⊢\exists x\exists y\exists z\exists w\phi \leftrightarrow \exists v\exists u\exists f\exists g\chi$$