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Mirrors > Home > MPE Home > Th. List > nfnbi | Structured version Visualization version GIF version |
Description: A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.) |
Ref | Expression |
---|---|
nfnbi | ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orcom 866 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑)) | |
2 | nf3 1787 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | |
3 | nf3 1787 | . . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑)) | |
4 | notnotb 317 | . . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
5 | 4 | albii 1820 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑) |
6 | 5 | orbi2i 909 | . . 3 ⊢ ((∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑)) |
7 | 3, 6 | bitr4i 280 | . 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑)) |
8 | 1, 2, 7 | 3bitr4i 305 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∨ wo 843 ∀wal 1535 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nfnt 1856 wl-sb8et 34804 |
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