MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stoic1a Structured version   Visualization version   GIF version

Theorem stoic1a 1688
Description: Stoic logic Thema 1 (part a).

The first thema of the four Stoic logic themata, in its basic form, was:

"When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/

We will represent thema 1 as two very similar rules stoic1a 1688 and stoic1b 1689 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)

Hypothesis
Ref Expression
stoic1.1 ((𝜑𝜓) → 𝜃)
Assertion
Ref Expression
stoic1a ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

Proof of Theorem stoic1a
StepHypRef Expression
1 stoic1.1 . . 3 ((𝜑𝜓) → 𝜃)
21ex 449 . 2 (𝜑 → (𝜓𝜃))
32con3dimp 456 1 ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383
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 196  df-an 385
This theorem is referenced by:  stoic1b  1689  posn  5110  frsn  5112  relimasn  5407  nssdmovg  6714  iblss  23377  midexlem  25387  colhp  25462  xaddeq0  28907  xrge0npcan  29025  unccur  32562  lindsenlbs  32574  itg2addnclem2  32632  dvasin  32666  ssnel  38227  icccncfext  38773  dirkercncflem1  38996  fourierdlem81  39080  fourierdlem97  39096  volico2  39531
  Copyright terms: Public domain W3C validator