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| Description: Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| nf3 | ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nf2 1784 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | |
| 2 | alnex 1780 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
| 3 | 2 | orbi2i 912 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | 
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∨ wo 847 ∀wal 1537 ∃wex 1778 Ⅎwnf 1782 | 
| 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 207 df-or 848 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: nf4 1786 nfim1 2198 dfnf5 4381 ab0orv 4382 | 
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