Theorem bj-nnf-cbvaliv

Description: The only DV conditions are those saying that y is a fresh variable used to construct ch . (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-cbvaliv.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-nnf-cbvaliv.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
	bj-nnf-cbvaliv.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
Assertion	bj-nnf-cbvaliv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbvaliv.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-nnf-cbvaliv.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
3		bj-nnf-cbvaliv.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
4		ax-5	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
5		ax-5	⊢ ( ∀ 𝑥 𝜓 → ∀ 𝑦 ∀ 𝑥 𝜓 )
6	1 2 3	bj-nnf-spim	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )
7	4 5 6	alrimdh	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )