Theorem bj-nnf-cbval

Description: Compared with cbvalv1 , this saves ax-12 . (Contributed by BJ, 4-Apr-2026)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbval.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-nnf-cbval.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-nnf-cbval.ps	⊢ ( 𝜑 → Ⅎ' 𝑦 𝜓 )
4		bj-nnf-cbval.ch	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
5		bj-nnf-cbval.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
6	5	biimpd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
7	1 2 3 4 6	bj-nnf-cbvali	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
8		equcomi	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
9	8 5	sylan2	⊢ ( ( 𝜑 ∧ 𝑦 = 𝑥 ) → ( 𝜓 ↔ 𝜒 ) )
10	9	biimprd	⊢ ( ( 𝜑 ∧ 𝑦 = 𝑥 ) → ( 𝜒 → 𝜓 ) )
11	2 1 4 3 10	bj-nnf-cbvali	⊢ ( 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
12	7 11	impbid	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )