Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  stoic4a GIF version

Theorem stoic4a 1409
 Description: Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)." We use 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1410 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)
Hypotheses
Ref Expression
stoic4a.1 ((𝜑𝜓) → 𝜒)
stoic4a.2 ((𝜒𝜑𝜃) → 𝜏)
Assertion
Ref Expression
stoic4a ((𝜑𝜓𝜃) → 𝜏)

Proof of Theorem stoic4a
StepHypRef Expression
1 stoic4a.1 . . 3 ((𝜑𝜓) → 𝜒)
213adant3 1002 . 2 ((𝜑𝜓𝜃) → 𝜒)
3 simp1 982 . 2 ((𝜑𝜓𝜃) → 𝜑)
4 simp3 984 . 2 ((𝜑𝜓𝜃) → 𝜃)
5 stoic4a.2 . 2 ((𝜒𝜑𝜃) → 𝜏)
62, 3, 4, 5syl3anc 1217 1 ((𝜑𝜓𝜃) → 𝜏)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116  df-3an 965 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator