Theorem bj-nnf-cbvaliv 37041

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

Hypotheses

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

Assertion

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

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝑥,𝑦

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

Proof of Theorem bj-nnf-cbvaliv

Step	Hyp	Ref	Expression
1		ax-5 1912	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		ax-5 1912	. 2 ⊢ (∀𝑥𝜓 → ∀𝑦∀𝑥𝜓)
3		bj-nnf-cbvaliv.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4		bj-nnf-cbvaliv.nf	. . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
5		bj-nnf-cbvaliv.is	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
6	3, 4, 5	bj-nnf-spim 37039	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
7	1, 2, 6	alrimdh 1865	1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  Ⅎ'wnnf 36991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-bj-nnf 36992

This theorem is referenced by: (None)