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Theorem pm5.6r 847
Description: Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If 𝜓 is decidable, the converse also holds (see pm5.6dc 846). (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
pm5.6r ((𝜑 → (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))

Proof of Theorem pm5.6r
StepHypRef Expression
1 pm2.53 651 . . 3 ((𝜓𝜒) → (¬ 𝜓𝜒))
21imim2i 12 . 2 ((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜓𝜒)))
32impd 246 1 ((𝜑 → (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ssundifim  3333
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