Theorem bj-nnf-cbvali 37814

Description: Compared with bj-nnf-cbvaliv 37811, replacing the DV condition on 𝑦, 𝜓 with the nonfreeness condition requires ax-11 2196. (Contributed by BJ, 4-Apr-2026.)

Hypotheses

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

Assertion

Ref	Expression
bj-nnf-cbvali	⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)

Proof of Theorem bj-nnf-cbvali

Step	Hyp	Ref	Expression
1		bj-nnf-cbvali.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		bj-nnf-cbvali.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		bj-nnf-cbvali.ps	. . . 4 ⊢ (𝜑 → Ⅎ'𝑦𝜓)
4	3	bj-nnfad 37764	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
5	2, 4	hbald 2207	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦∀𝑥𝜓))
6		bj-nnf-cbvali.ch	. . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
7		bj-nnf-cbvali.is	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
8	2, 6, 7	bj-nnf-spim 37809	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
9	1, 5, 8	bj-alrimd 37628	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  Ⅎ'wnnf 37761

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-6 2001  ax-11 2196

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-bj-nnf 37762

This theorem is used by:  bj-nnf-cbval  37815