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Theorem pm2.18 125
 Description: Clavius's law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law pm2.01 181. (Contributed by NM, 29-Dec-1992.)
Assertion
Ref Expression
pm2.18 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem pm2.18
StepHypRef Expression
1 pm2.21 121 . . . 4 𝜑 → (𝜑 → ¬ (¬ 𝜑𝜑)))
21a2i 14 . . 3 ((¬ 𝜑𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑𝜑)))
32con4d 115 . 2 ((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑))
43pm2.43i 52 1 ((¬ 𝜑𝜑) → 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  126  pm2.18d  127  notnotr  128  pm4.81  384  sumdmdlem2  29850  axc11n11r  33262
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