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

Theorem stoic1a 1768
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 1768 and stoic1b 1769 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 412 . 2 (𝜑 → (𝜓𝜃))
32con3dimp 408 1 ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395
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 207  df-an 396
This theorem is referenced by:  stoic1b  1769  posn  5773  frsn  5775  relimasn  6104  nssdmovg  7614  iblss  25854  midexlem  28714  colhp  28792  clwwlknon0  30121  xaddeq0  32763  xrge0npcan  33007  madjusmdetlem2  33788  unccur  37589  lindsenlbs  37601  itg2addnclem2  37658  dvasin  37690  ssnel  44980  icccncfext  45842  dirkercncflem1  46058  fourierdlem81  46142  fourierdlem97  46158  prsal  46273  volico2  46596
  Copyright terms: Public domain W3C validator