Metamath Proof Explorer

Theorem stoic3

Description: Stoic logic Thema 3. Statement T3 of Bobzien p. 116-117 discusses Stoic logic Thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external assumption, another follows, then that other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019)

Ref Expression
Hypotheses stoic3.1 ${⊢}\left({\phi }\wedge {\psi }\right)\to {\chi }$
stoic3.2 ${⊢}\left({\chi }\wedge {\theta }\right)\to {\tau }$
Assertion stoic3 ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\tau }$

Proof

Step Hyp Ref Expression
1 stoic3.1 ${⊢}\left({\phi }\wedge {\psi }\right)\to {\chi }$
2 stoic3.2 ${⊢}\left({\chi }\wedge {\theta }\right)\to {\tau }$
3 1 2 sylan ${⊢}\left(\left({\phi }\wedge {\psi }\right)\wedge {\theta }\right)\to {\tau }$
4 3 3impa ${⊢}\left({\phi }\wedge {\psi }\wedge {\theta }\right)\to {\tau }$