Theorem bj-nnftht 37409

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 2222 (modal T), as in bj-nnfbi 37413. (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 37404	. 2 ⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))
2	1	biimpa 482	1 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀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-nnfth  37410