Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege18 Structured version   Visualization version   GIF version

Theorem frege18 40042
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 40024 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜃 → (𝜓𝜒))))
2 frege16 40040 . 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 40014  ax-frege2 40015  ax-frege8 40033
This theorem is referenced by:  frege19  40048  frege23  40049  frege20  40052  frege51  40079  frege64a  40106  frege64b  40133  frege64c  40151  frege82  40169
  Copyright terms: Public domain W3C validator