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| Mirrors > Home > MPE Home > Th. List > nbbn | Structured version Visualization version GIF version | ||
| Description: Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.) |
| Ref | Expression |
|---|---|
| nbbn | ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm5.18 383 | . 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (¬ 𝜑 ↔ ¬ 𝜓)) | |
| 2 | notbi 321 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) | |
| 3 | 1, 2 | xchbinxr 337 | 1 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 208 |
| 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 |
| This theorem is referenced by: biass 387 pclem6 1039 xorass 1535 hadbi 1618 canth 7350 qextltlem 13205 onint1 36806 wl-df3xor2 37960 wl-3xornot1 37971 wl-2xor 37974 notbinot1 38575 notbinot2 38579 onsupmaxb 43813 |
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