Theorem stoic1a 1476

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 1476 and stoic1b 1477 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	|- ( ( ph /\ ps ) -> th )

Assertion

Ref	Expression
stoic1a	|- ( ( ph /\ -. th ) -> -. ps )

Proof of Theorem stoic1a

Step	Hyp	Ref	Expression
1		stoic1.1	. . 3 |- ( ( ph /\ ps ) -> th )
2	1	ex 115	. 2 |- ( ph -> ( ps -> th ) )
3	2	con3dimp 644	1 |- ( ( ph /\ -. th ) -> -. ps )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This theorem is used by:  stoic1b  1477  bassetsnn  13409