Theorem bj-nnftht 37182

Description: A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2217 (modal T), as in bj-nnfbi 37186. (Contributed by BJ, 28-Jul-2023.)

Assertion

Ref	Expression
bj-nnftht	⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

Proof of Theorem bj-nnftht

Step	Hyp	Ref	Expression
1		bj-alnnf2 37177	. 2 ⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))
2	1	biimpa 480	1 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557  Ⅎ'wnnf 37165

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 209  df-an 400  df-bj-nnf 37166

This theorem is referenced by:  bj-nnfth  37183