Theorem bj-alnnf2 36975

Description: If a proposition holds, then it holds for all values of a given variable if and only if it does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)

Assertion

Ref	Expression
bj-alnnf2	⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))

Proof of Theorem bj-alnnf2

Step	Hyp	Ref	Expression
1		bj-alnnf 36974	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))
2	1	pm5.74ri 272	1 ⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  Ⅎ'wnnf 36963

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 207  df-an 396  df-bj-nnf 36964

This theorem is referenced by:  bj-nnftht  36980