Theorem bj-cbvalimdv

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

	Ref	Expression
Hypotheses	bj-cbvalimdv.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-cbvalimdv.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-cbvalimdv.nfth	⊢ ( 𝜑 → ( ∃ 𝑥 𝜃 → 𝜃 ) )
	bj-cbvalimdv.denote	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 )
	bj-cbvalimdv.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
Assertion	bj-cbvalimdv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜒 → ∀ 𝑦 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvalimdv.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-cbvalimdv.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-cbvalimdv.nfth	⊢ ( 𝜑 → ( ∃ 𝑥 𝜃 → 𝜃 ) )
4		bj-cbvalimdv.denote	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜓 )
5		bj-cbvalimdv.maj	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
6		ax5d	⊢ ( 𝜑 → ( ∀ 𝑥 𝜒 → ∀ 𝑦 ∀ 𝑥 𝜒 ) )
7	1 2 6 3 4 5	bj-cbvalimdlem	⊢ ( 𝜑 → ( ∀ 𝑥 𝜒 → ∀ 𝑦 𝜃 ) )