Theorem bj-nnftht 34089

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 2181 (modal T), as in bj-nnfbi 34076. (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		ax-1 6	. . 3 ⊢ (𝜑 → (∃𝑥𝜑 → 𝜑))
2		ax-1 6	. . 3 ⊢ (∀𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	1, 2	anim12i 614	. 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
4		df-bj-nnf 34075	. 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5	3, 4	sylibr 236	1 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34074

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 399  df-bj-nnf 34075

This theorem is referenced by:  bj-nnfth  34090