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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 2942 | . 2 ⊢ ¬ 𝐴 = 𝐵 |
3 | nemtbir.2 | . 2 ⊢ (𝜑 ↔ 𝐴 = 𝐵) | |
4 | 2, 3 | mtbir 323 | 1 ⊢ ¬ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ≠ wne 2940 |
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 2941 |
This theorem is referenced by: opthwiener 5472 opthprc 5697 ord2eln012 8444 snnen2oOLD 9174 0sdom1dom 9185 cfpwsdom 10525 fprodn0f 15879 m1exp1 16263 pmtrsn 19306 gzrngunitlem 20878 logbmpt 26154 sltval2 27020 sltsolem1 27039 nolt02o 27059 ex-id 29420 ex-mod 29435 coss0 36987 ensucne0 41889 clsk1indlem4 42404 clsk1indlem1 42405 etransc 44610 |
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