![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2932 | . 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 1534 ≠ wne 2930 |
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 2931 |
This theorem is referenced by: opthwiener 5520 opthprc 5746 ord2eln012 8527 snnen2oOLD 9261 0sdom1dom 9272 cfpwsdom 10627 fprodn0f 15993 m1exp1 16378 pmtrsn 19517 gzrngunitlem 21429 logbmpt 26816 sltval2 27686 sltsolem1 27705 nolt02o 27725 ex-id 30367 ex-mod 30382 coss0 38177 ensucne0 43196 clsk1indlem4 43711 clsk1indlem1 43712 etransc 45904 |
Copyright terms: Public domain | W3C validator |