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  5709  frsn  5711  relimasn  6043  nssdmovg  7540  iblss  25764  midexlem  28745  colhp  28823  clwwlknon0  30149  xaddeq0  32812  xrge0npcan  33081  elrgspnsubrunlem2  33309  esplyfval3  33709  constrinvcl  33909  madjusmdetlem2  33964  onvf1od  35280  unccur  37773  lindsenlbs  37785  itg2addnclem2  37842  dvasin  37874  ssnel  45325  icccncfext  46168  dirkercncflem1  46384  fourierdlem81  46468  fourierdlem97  46484  prsal  46599  volico2  46922