Theorem bj-nnf-cbvali

Description: Compared with bj-nnf-cbvaliv , replacing the DV condition on y , ps with the nonfreeness condition requires ax-11 . (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-cbvali.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-nnf-cbvali.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-nnf-cbvali.ps	⊢ ( 𝜑 → Ⅎ' 𝑦 𝜓 )
	bj-nnf-cbvali.ch	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
	bj-nnf-cbvali.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
Assertion	bj-nnf-cbvali	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbvali.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-nnf-cbvali.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-nnf-cbvali.ps	⊢ ( 𝜑 → Ⅎ' 𝑦 𝜓 )
4		bj-nnf-cbvali.ch	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
5		bj-nnf-cbvali.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
6	3	bj-nnfad	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
7	1 6	hbald	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 ∀ 𝑥 𝜓 ) )
8	1 4 5	bj-nnf-spim	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )
9	2 7 8	bj-alrimd	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )