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Theorem pm4.82 964
Description: Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
Assertion
Ref Expression
pm4.82 (((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

Proof of Theorem pm4.82
StepHypRef Expression
1 pm2.65 182 . . 3 ((𝜑𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
21imp 443 . 2 (((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜑)
3 pm2.21 118 . . 3 𝜑 → (𝜑𝜓))
4 pm2.21 118 . . 3 𝜑 → (𝜑 → ¬ 𝜓))
53, 4jca 552 . 2 𝜑 → ((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)))
62, 5impbii 197 1 (((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195  df-an 384
This theorem is referenced by:  alimp-no-surprise  42292
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