# Metamath Proof Explorer

## Theorem stoic4a

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 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019)

Ref Expression
Hypotheses stoic4a.1 ${⊢}\left({\phi }\wedge {\psi }\right)\to {\chi }$
stoic4a.2 ${⊢}\left({\chi }\wedge {\phi }\wedge {\theta }\right)\to {\tau }$
Assertion stoic4a ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\tau }$

### Proof

Step Hyp Ref Expression
1 stoic4a.1 ${⊢}\left({\phi }\wedge {\psi }\right)\to {\chi }$
2 stoic4a.2 ${⊢}\left({\chi }\wedge {\phi }\wedge {\theta }\right)\to {\tau }$
3 1 3adant3 ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\chi }$
4 simp1 ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\phi }$
5 simp3 ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\theta }$
6 3 4 5 2 syl3anc ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\tau }$