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	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-cbveximdv.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-cbveximdv.nfth	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑦 𝜒 ) )
	bj-cbveximdv.denote	⊢ ( 𝜑 → ∀ 𝑥 ∃ 𝑦 𝜓 )
	bj-cbveximdv.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
Assertion	bj-cbveximdv	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → ∃ 𝑦 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbveximdv.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-cbveximdv.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-cbveximdv.nfth	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑦 𝜒 ) )
4		bj-cbveximdv.denote	⊢ ( 𝜑 → ∀ 𝑥 ∃ 𝑦 𝜓 )
5		bj-cbveximdv.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
6		ax5e	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜃 → ∃ 𝑦 𝜃 )
7	6	a1i	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 𝜃 → ∃ 𝑦 𝜃 ) )
8	1 2 3 7 4 5	bj-cbveximdlem	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → ∃ 𝑦 𝜃 ) )