Theorem stoic1a 1804

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 1804 and stoic1b 1805 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 418	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
3	2	con3dimp 414	1 ⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  stoic1b  1805  posn  5736  frsn  5738  relimasn  6082  nssdmovg  7603  lindsenlbs  22157  iblss  26125  midexlem  29164  colhp  29248  plngrotlem1  29265  prlngplngtr  29437  clwwlknon0  30684  xaddeq0  33345  xrge0npcan  33581  elrgspnsubrunlem2  33809  drnglring  34024  esplyfval3  34204  constrinvcl  34405  madjusmdetlem2  34460  onvf1od  35886  unccur  38527  itg2addnclem2  38591  dvasin  38623  ssnel  46060  icccncfext  46897  dirkercncflem1  47113  fourierdlem81  47197  fourierdlem97  47213  prsal  47328  volico2  47651  indprmfz  48715