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Mirrors > Home > MPE Home > Th. List > pm2.65d | Structured version Visualization version GIF version |
Description: Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) |
Ref | Expression |
---|---|
pm2.65d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
pm2.65d.2 | ⊢ (𝜑 → (𝜓 → ¬ 𝜒)) |
Ref | Expression |
---|---|
pm2.65d | ⊢ (𝜑 → ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.65d.2 | . . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒)) | |
2 | pm2.65d.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | nsyld 153 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓)) |
4 | 3 | pm2.01d 180 | 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: mtod 188 pm2.65da 598 unxpdomlem2 8050 cardlim 8681 winainflem 9394 winalim2 9397 discr 12863 sqrmo 13840 vdwnnlem3 15539 nmlno0lem 27032 nmlnop0iALT 28238 iooelexlt 32386 |
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