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Mirrors > Home > MPE Home > Th. List > pm2.18 | Structured version Visualization version GIF version |
Description: Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. See also pm2.01 179. (Contributed by NM, 29-Dec-1992.) |
Ref | Expression |
---|---|
pm2.18 | ⊢ ((¬ 𝜑 → 𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 119 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → ¬ (¬ 𝜑 → 𝜑))) | |
2 | 1 | a2i 14 | . . 3 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑))) |
3 | 2 | con4d 113 | . 2 ⊢ ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)) |
4 | 3 | pm2.43i 50 | 1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑) |
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: pm2.18i 122 pm2.18d 123 pm4.81 380 sumdmdlem2 28662 pm4.81ALT 31716 axc11n11r 31860 |
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