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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfnt | Structured version Visualization version GIF version | ||
| Description: A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 36748). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1855. (Contributed by BJ, 28-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| bj-nnfnt | ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eximal 1781 | . . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
| 2 | alimex 1830 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (∃𝑥 ¬ 𝜑 → ¬ 𝜑)) | |
| 3 | 1, 2 | anbi12ci 629 | . 2 ⊢ (((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))) | 
| 4 | df-bj-nnf 36726 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
| 5 | df-bj-nnf 36726 | . 2 ⊢ (Ⅎ'𝑥 ¬ 𝜑 ↔ ((∃𝑥 ¬ 𝜑 → ¬ 𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 Ⅎ'wnnf 36725 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-bj-nnf 36726 | 
| This theorem is referenced by: bj-nnfnth 36745 bj-equsexvwd 36783 | 
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