Theorem stoic1a 1773

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 1773 and stoic1b 1774 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  1774  posn  5700  frsn  5702  relimasn  6033  nssdmovg  7528  iblss  25733  midexlem  28670  colhp  28748  clwwlknon0  30073  xaddeq0  32736  xrge0npcan  33001  elrgspnsubrunlem2  33215  constrinvcl  33786  madjusmdetlem2  33841  onvf1od  35151  unccur  37642  lindsenlbs  37654  itg2addnclem2  37711  dvasin  37743  ssnel  45139  icccncfext  45984  dirkercncflem1  46200  fourierdlem81  46284  fourierdlem97  46300  prsal  46415  volico2  46738