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Mirrors > Home > MPE Home > Th. List > necon2abid | Structured version Visualization version GIF version |
Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
Ref | Expression |
---|---|
necon2abid.1 | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓)) |
Ref | Expression |
---|---|
necon2abid | ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 315 | . 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
2 | necon2abid.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓)) | |
3 | 2 | necon3abid 2975 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓)) |
4 | 1, 3 | bitr4id 290 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1537 ≠ wne 2938 |
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-ne 2939 |
This theorem is referenced by: sossfld 6208 funeldmb 7379 fin23lem24 10360 isf32lem4 10394 sqgt0sr 11144 leltne 11348 xrleltne 13184 xrltne 13202 ge0nemnf 13212 xlt2add 13299 supxrbnd 13367 supxrre2 13370 ioopnfsup 13901 icopnfsup 13902 xblpnfps 24421 xblpnf 24422 nmoreltpnf 30798 nmopreltpnf 31898 elprneb 46979 |
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