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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 179 pm2.18da 800 oplem1 1056 axc11n 2429 weniso 7374 infssuni 9384 ordtypelem10 9565 oismo 9578 rankval3b 9864 grur1 10858 sqeqd 15202 hausflimi 24004 minveclem4 25480 ovolunnul 25549 vitali 25662 itg2mono 25803 frgrncvvdeqlem8 30335 minvecolem4 30909 contrd 38084 fppr2odd 47656 |
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