Theorem bj-alnnf 37224

Description: In deduction-style proofs, it is equivalent to assert that the context holds for all values of a variable, or that is does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)

Assertion

Ref	Expression
bj-alnnf	⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))

Proof of Theorem bj-alnnf

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (𝜑 → (∃𝑥𝜑 → 𝜑))
2	1	biantrur 539	. . 3 ⊢ ((𝜑 → (𝜑 → ∀𝑥𝜑)) ↔ ((𝜑 → (∃𝑥𝜑 → 𝜑)) ∧ (𝜑 → (𝜑 → ∀𝑥𝜑))))
3		pm5.4 392	. . 3 ⊢ ((𝜑 → (𝜑 → ∀𝑥𝜑)) ↔ (𝜑 → ∀𝑥𝜑))
4		pm4.76 527	. . 3 ⊢ (((𝜑 → (∃𝑥𝜑 → 𝜑)) ∧ (𝜑 → (𝜑 → ∀𝑥𝜑))) ↔ (𝜑 → ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))))
5	2, 3, 4	3bitr3i 304	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))))
6		df-bj-nnf 37214	. . 3 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
7	6	imbi2i 339	. 2 ⊢ ((𝜑 → Ⅎ'𝑥𝜑) ↔ (𝜑 → ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))))
8	5, 7	bitr4i 281	1 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802  Ⅎ'wnnf 37213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-bj-nnf 37214

This theorem is referenced by:  bj-alnnf2  37225