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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnftht | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bj-nnftht | ⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (𝜑 → (∃𝑥𝜑 → 𝜑)) | |
2 | ax-1 6 | . . 3 ⊢ (∀𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | anim12i 613 | . 2 ⊢ ((𝜑 ∧ ∀𝑥𝜑) → ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) |
4 | df-bj-nnf 34906 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
5 | 3, 4 | sylibr 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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