Theorem bj-alnnf2 37404

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 37403	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))
2	1	pm5.74ri 275	1 ⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎ'wnnf 37392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-bj-nnf 37393

This theorem is used by:  bj-nnftht  37409