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Theorem frege11 40437
Description: Elimination of a nested antecedent as a partial converse of ja 189. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege11 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Proof of Theorem frege11
StepHypRef Expression
1 ax-frege1 40413 . 2 (𝜓 → (𝜑𝜓))
2 frege9 40435 . 2 ((𝜓 → (𝜑𝜓)) → (((𝜑𝜓) → 𝜒) → (𝜓𝜒)))
31, 2ax-mp 5 1 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-frege1 40413  ax-frege2 40414  ax-frege8 40432
This theorem is referenced by:  frege112  40598
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