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Mirrors > Home > MPE Home > Th. List > mtbii | Structured version Visualization version GIF version |
Description: An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.) |
Ref | Expression |
---|---|
mtbii.min | ⊢ ¬ 𝜓 |
mtbii.maj | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mtbii | ⊢ (𝜑 → ¬ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtbii.min | . 2 ⊢ ¬ 𝜓 | |
2 | mtbii.maj | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimprd 247 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
4 | 1, 3 | mtoi 198 | 1 ⊢ (𝜑 → ¬ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 |
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 |
This theorem is referenced by: limom 7871 omopthlem2 8659 fineqv 9263 nd3 10584 axunndlem1 10590 axregndlem1 10597 axregndlem2 10598 axregnd 10599 axacndlem5 10606 canthp1lem2 10648 alephgch 10669 inatsk 10773 addnidpi 10896 indpi 10902 archnq 10975 fsumsplit 15687 sumsplit 15714 geoisum1c 15826 fprodm1 15911 m1dvdsndvds 16731 gexdvds 19452 chtub 26715 nolt02o 27198 nogt01o 27199 wlkp1lem6 28966 avril1 29747 ballotlemi1 33532 ballotlemii 33533 distel 34806 onsucsuccmpi 35376 bj-inftyexpitaudisj 36134 bj-inftyexpidisj 36139 poimirlem28 36564 poimirlem32 36568 n0eldmqseq 37567 lcmineqlem23 40964 nvelim 45879 0nodd 46628 2nodd 46630 |
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