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Mirrors > Home > MPE Home > Th. List > nf3 | Structured version Visualization version GIF version |
Description: Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.) |
Ref | Expression |
---|---|
nf3 | ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf2 1780 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | |
2 | alnex 1776 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
3 | 2 | orbi2i 909 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∨ wo 843 ∀wal 1529 ∃wex 1774 Ⅎwnf 1778 |
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-or 844 df-ex 1775 df-nf 1779 |
This theorem is referenced by: nf4 1782 nfnbi 1849 nfim1 2192 dfnf5 4332 |
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