Theorem stoic1a 1774

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 1774 and stoic1b 1775 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	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Assertion

Ref	Expression
stoic1a	⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

Proof of Theorem stoic1a

Step	Hyp	Ref	Expression
1		stoic1.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
2	1	ex 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
3	2	con3dimp 408	1 ⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

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  1775  posn  5717  frsn  5719  relimasn  6050  nssdmovg  7549  iblss  25772  midexlem  28760  colhp  28838  clwwlknon0  30163  xaddeq0  32826  xrge0npcan  33080  elrgspnsubrunlem2  33309  esplyfval3  33716  constrinvcl  33917  madjusmdetlem2  33972  onvf1od  35289  unccur  37924  lindsenlbs  37936  itg2addnclem2  37993  dvasin  38025  ssnel  45474  icccncfext  46315  dirkercncflem1  46531  fourierdlem81  46615  fourierdlem97  46631  prsal  46746  volico2  47069  indprmfz  48093