Theorem stoic1a 1772

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 1772 and stoic1b 1773 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  1773  posn  5724  frsn  5726  relimasn  6056  nssdmovg  7571  iblss  25706  midexlem  28619  colhp  28697  clwwlknon0  30022  xaddeq0  32676  xrge0npcan  32961  elrgspnsubrunlem2  33199  constrinvcl  33763  madjusmdetlem2  33818  onvf1od  35094  unccur  37597  lindsenlbs  37609  itg2addnclem2  37666  dvasin  37698  ssnel  45037  icccncfext  45885  dirkercncflem1  46101  fourierdlem81  46185  fourierdlem97  46201  prsal  46316  volico2  46639