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Theorem bj-nnftht 34923
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 2176 (modal T), as in bj-nnfbi 34907. (Contributed by BJ, 28-Jul-2023.)
Assertion
Ref Expression
bj-nnftht ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)

Proof of Theorem bj-nnftht
StepHypRef Expression
1 ax-1 6 . . 3 (𝜑 → (∃𝑥𝜑𝜑))
2 ax-1 6 . . 3 (∀𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
31, 2anim12i 613 . 2 ((𝜑 ∧ ∀𝑥𝜑) → ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
4 df-bj-nnf 34906 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
53, 4sylibr 233 1 ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537  wex 1782  Ⅎ'wnnf 34905
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 206  df-an 397  df-bj-nnf 34906
This theorem is referenced by:  bj-nnfth  34924
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