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Theorem pm2.65 584
 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 774, 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 35 . . . 4 (φ → ((φ → ¬ ψ) → ¬ ψ))
21con2d 554 . . 3 (φ → (ψ → ¬ (φ → ¬ ψ)))
32a2i 11 . 2 ((φψ) → (φ → ¬ (φ → ¬ ψ)))
43con2d 554 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 544  ax-in2 545 This theorem is referenced by:  pm4.82  856
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