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Theorem pm2.18 128
Description: Clavius 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 pm2.01 192. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17-Nov-2023.)
Assertion
Ref Expression
pm2.18 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem pm2.18
StepHypRef Expression
1 id 22 . 2 ((¬ 𝜑𝜑) → (¬ 𝜑𝜑))
21pm2.18d 127 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  131  notnotr  132  pm4.81  397  sumdmdlem2  30198  axc11n11r  34042
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