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Theorem frege47 38462
 Description: Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege47 ((¬ 𝜑𝜓) → ((𝜓𝜒) → ((𝜑𝜒) → 𝜒)))

Proof of Theorem frege47
StepHypRef Expression
1 frege46 38461 . 2 ((¬ 𝜑𝜒) → ((𝜑𝜒) → 𝜒))
2 frege21 38438 . 2 (((¬ 𝜑𝜒) → ((𝜑𝜒) → 𝜒)) → ((¬ 𝜑𝜓) → ((𝜓𝜒) → ((𝜑𝜒) → 𝜒))))
31, 2ax-mp 5 1 ((¬ 𝜑𝜓) → ((𝜓𝜒) → ((𝜑𝜒) → 𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege28 38441  ax-frege31 38445  ax-frege41 38456 This theorem is referenced by:  frege48  38463  frege49  38464
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