Theorem bj-nnf-cbval 37138

Description: Compared with cbvalv1 2351, this saves ax-12 2191. (Contributed by BJ, 4-Apr-2026.)

Hypotheses

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

Assertion

Ref	Expression
bj-nnf-cbval	⊢ (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-cbval

Step	Hyp	Ref	Expression
1		bj-nnf-cbval.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-nnf-cbval.nf1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-nnf-cbval.ps	. . 3 ⊢ (𝜑 → Ⅎ'𝑦𝜓)
4		bj-nnf-cbval.ch	. . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
5		bj-nnf-cbval.is	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
6	5	biimpd 231	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
7	1, 2, 3, 4, 6	bj-nnf-cbvali 37137	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
8		equcomi 2025	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
9	8, 5	sylan2 600	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑥) → (𝜓 ↔ 𝜒))
10	9	biimprd 250	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑥) → (𝜒 → 𝜓))
11	2, 1, 4, 3, 10	bj-nnf-cbvali 37137	. 2 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
12	7, 11	impbid 214	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546  Ⅎ'wnnf 37084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-11 2170

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-bj-nnf 37085

This theorem is referenced by: (None)