Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  stoic2a Structured version   Visualization version   GIF version

Theorem stoic2a 1697
 Description: Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as 𝜑, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1698 for the version with the phrase "or both". We already have this rule as syldan 487, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
Hypotheses
Ref Expression
stoic2a.1 ((𝜑𝜓) → 𝜒)
stoic2a.2 ((𝜑𝜒) → 𝜃)
Assertion
Ref Expression
stoic2a ((𝜑𝜓) → 𝜃)

Proof of Theorem stoic2a
StepHypRef Expression
1 stoic2a.1 . 2 ((𝜑𝜓) → 𝜒)
2 stoic2a.2 . 2 ((𝜑𝜒) → 𝜃)
31, 2syldan 487 1 ((𝜑𝜓) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-an 386 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator