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Theorem pm2.65 595
Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here 𝜑, derive a contradiction, and therefore conclude ¬ 𝜑, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume ¬ 𝜑, derive a contradiction, and conclude 𝜑, such as condandc 786, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
Assertion
Ref Expression
pm2.65 ((𝜑𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))

Proof of Theorem pm2.65
StepHypRef Expression
1 pm2.27 39 . . . 4 (𝜑 → ((𝜑 → ¬ 𝜓) → ¬ 𝜓))
21con2d 564 . . 3 (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
32a2i 11 . 2 ((𝜑𝜓) → (𝜑 → ¬ (𝜑 → ¬ 𝜓)))
43con2d 564 1 ((𝜑𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 554  ax-in2 555
This theorem is referenced by:  pm4.82  868
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