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Mirrors > Home > MPE Home > Th. List > pm2.18d | Structured version Visualization version GIF version |
Description: Deduction form of the Clavius law pm2.18 128. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Revised to shorten pm2.18 128. (Revised by Wolf Lammen, 17-Nov-2023.) |
Ref | Expression |
---|---|
pm2.18d.1 | ⊢ (𝜑 → (¬ 𝜓 → 𝜓)) |
Ref | Expression |
---|---|
pm2.18d | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | pm2.18d.1 | . . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜓)) | |
3 | pm2.21 123 | . . 3 ⊢ (¬ 𝜓 → (𝜓 → ¬ 𝜑)) | |
4 | 2, 3 | sylcom 30 | . 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜑)) |
5 | 1, 4 | mt4d 117 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: pm2.18 128 pm2.61d 182 pm2.18da 799 oplem1 1052 axc11n 2437 weniso 7086 infssuni 8799 ordtypelem10 8975 oismo 8988 rankval3b 9239 grur1 10231 sqeqd 14517 hausflimi 22585 minveclem4 24036 ovolunnul 24104 vitali 24217 itg2mono 24357 frgrncvvdeqlem8 28091 minvecolem4 28663 contrd 35535 fppr2odd 44249 |
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