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Mirrors > Home > MPE Home > Th. List > nemtbir | Structured version Visualization version GIF version |
Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.) |
Ref | Expression |
---|---|
nemtbir.1 | ⊢ 𝐴 ≠ 𝐵 |
nemtbir.2 | ⊢ (𝜑 ↔ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nemtbir | ⊢ ¬ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nemtbir.1 | . . 3 ⊢ 𝐴 ≠ 𝐵 | |
2 | 1 | neii 2944 | . 2 ⊢ ¬ 𝐴 = 𝐵 |
3 | nemtbir.2 | . 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵) | |
4 | 2, 3 | mtbir 322 | 1 ⊢ ¬ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1539 ≠ wne 2942 |
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 206 df-ne 2943 |
This theorem is referenced by: opthwiener 5422 opthprc 5642 snnen2o 8903 cfpwsdom 10271 fprodn0f 15629 m1exp1 16013 pmtrsn 19042 gzrngunitlem 20575 logbmpt 25843 ex-id 28699 ex-mod 28714 sltval2 33786 sltsolem1 33805 nolt02o 33825 coss0 36524 ensucne0 41034 clsk1indlem4 41543 clsk1indlem1 41544 etransc 43714 |
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