Theorem bj-cbveximdv

Description: A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbveximdv.nf0	$$⊢\phi \to \forall x\phi$$
	bj-cbveximdv.nf1	$$⊢\phi \to \forall y\phi$$
	bj-cbveximdv.nfth	$$⊢\phi \to \left(\chi \to \forall y\chi \right)$$
	bj-cbveximdv.denote	$$⊢\phi \to \forall x\exists y\psi$$
	bj-cbveximdv.maj	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \to \theta \right)$$
Assertion	bj-cbveximdv	$$⊢\phi \to \left(\exists x\chi \to \exists y\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbveximdv.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-cbveximdv.nf1	$$⊢\phi \to \forall y\phi$$
3		bj-cbveximdv.nfth	$$⊢\phi \to \left(\chi \to \forall y\chi \right)$$
4		bj-cbveximdv.denote	$$⊢\phi \to \forall x\exists y\psi$$
5		bj-cbveximdv.maj	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \to \theta \right)$$
6		ax5e	$$⊢\exists x\exists y\theta \to \exists y\theta$$
7	6	a1i	$$⊢\phi \to \left(\exists x\exists y\theta \to \exists y\theta \right)$$
8	1 2 3 7 4 5	bj-cbveximdlem	$$⊢\phi \to \left(\exists x\chi \to \exists y\theta \right)$$