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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.) |
Ref | Expression |
---|---|
nbbn | ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor3 387 | . 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)) | |
2 | con2bi 357 | . 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)) | |
3 | bicom 225 | . 2 ⊢ ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | 3bitrri 301 | 1 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 |
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 210 |
This theorem is referenced by: biass 389 pclem6 1023 xorass 1507 hadbi 1599 canth 7090 qextltlem 12583 onint1 33910 wl-df3xor2 34886 wl-3xornot1 34897 wl-2xor 34900 notbinot1 35517 notbinot2 35521 |
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