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Theorem frege18 36931
Description: Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege18 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒))))

Proof of Theorem frege18
StepHypRef Expression
1 frege5 36913 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜃 → (𝜓𝜒))))
2 frege16 36929 . 2 (((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜃 → (𝜓𝜒)))) → ((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒)))))
31, 2ax-mp 5 1 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-frege1 36903  ax-frege2 36904  ax-frege8 36922
This theorem is referenced by:  frege19  36937  frege23  36938  frege20  36941  frege51  36968  frege64a  36995  frege64b  37022  frege64c  37040  frege82  37058
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