Theorem stoic4a 1443

Description: Stoic logic Thema 4 version a.

Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use th to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1444 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

Hypotheses

Ref	Expression
stoic4a.1	|- ( ( ph /\ ps ) -> ch )
stoic4a.2	|- ( ( ch /\ ph /\ th ) -> ta )

Assertion

Ref	Expression
stoic4a	|- ( ( ph /\ ps /\ th ) -> ta )

Proof of Theorem stoic4a

Step	Hyp	Ref	Expression
1		stoic4a.1	. . 3 |- ( ( ph /\ ps ) -> ch )
2	1	3adant3 1019	. 2 |- ( ( ph /\ ps /\ th ) -> ch )
3		simp1 999	. 2 |- ( ( ph /\ ps /\ th ) -> ph )
4		simp3 1001	. 2 |- ( ( ph /\ ps /\ th ) -> th )
5		stoic4a.2	. 2 |- ( ( ch /\ ph /\ th ) -> ta )
6	2, 3, 4, 5	syl3anc 1249	1 |- ( ( ph /\ ps /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by: (None)