 Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege34 Structured version   Visualization version   GIF version

Theorem frege34 38971
 Description: If as a conseqence of the occurence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurence of the obstacle 𝜓 can be inferred. Closed form of con1d 142. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege34 ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))

Proof of Theorem frege34
StepHypRef Expression
1 frege33 38970 . 2 ((¬ 𝜓𝜒) → (¬ 𝜒𝜓))
2 frege5 38934 . 2 (((¬ 𝜓𝜒) → (¬ 𝜒𝜓)) → ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓))))
31, 2ax-mp 5 1 ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-frege1 38924  ax-frege2 38925  ax-frege28 38964  ax-frege31 38968 This theorem is referenced by:  frege35  38972  frege36  38973
 Copyright terms: Public domain W3C validator