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

Theorem stoic2a 1689
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, 𝜓𝜒; in Metamath we will represent that construct as 𝜑𝜓𝜒. This version a is without the phrase "or both"; see stoic2b 1690 for the version with the phrase "or both". We already have this rule as syldan 485, 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 485 1 ((𝜑𝜓) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382
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 195  df-an 384
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator