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Theorem frege47 42028
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 42027 . 2 ((¬ 𝜑𝜒) → ((𝜑𝜒) → 𝜒))
2 frege21 42004 . 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 41967  ax-frege2 41968  ax-frege8 41986  ax-frege28 42007  ax-frege31 42011  ax-frege41 42022
This theorem is referenced by:  frege48  42029  frege49  42030
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