Theorem stoic1a

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 and stoic1b to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019) (Proof shortened by Wolf Lammen, 21-May-2020)

	Ref	Expression
Hypothesis	stoic1.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
Assertion	stoic1a	⊢ ( ( 𝜑 ∧ ¬ 𝜃 ) → ¬ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		stoic1.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
2	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3	2	con3dimp	⊢ ( ( 𝜑 ∧ ¬ 𝜃 ) → ¬ 𝜓 )