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  5705  frsn  5707  relimasn  6036  nssdmovg  7531  iblss  25704  midexlem  28637  colhp  28715  clwwlknon0  30037  xaddeq0  32697  xrge0npcan  32975  elrgspnsubrunlem2  33189  constrinvcl  33746  madjusmdetlem2  33801  onvf1od  35090  unccur  37593  lindsenlbs  37605  itg2addnclem2  37662  dvasin  37694  ssnel  45031  icccncfext  45878  dirkercncflem1  46094  fourierdlem81  46178  fourierdlem97  46194  prsal  46309  volico2  46632