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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 318 | . 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
2 | necon2abid.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓)) | |
3 | 2 | necon3abid 3023 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓)) |
4 | 1, 3 | bitr4id 293 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ≠ wne 2987 |
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 df-ne 2988 |
This theorem is referenced by: sossfld 6010 fin23lem24 9733 isf32lem4 9767 sqgt0sr 10517 leltne 10719 xrleltne 12526 xrltne 12544 ge0nemnf 12554 xlt2add 12641 supxrbnd 12709 supxrre2 12712 ioopnfsup 13227 icopnfsup 13228 xblpnfps 23002 xblpnf 23003 nmoreltpnf 28552 nmopreltpnf 29652 funeldmb 33119 elprneb 43621 |
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