Metamath Proof Explorer


Theorem syl3an

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004)

Ref Expression
Hypotheses syl3an.1
|- ( ph -> ps )
syl3an.2
|- ( ch -> th )
syl3an.3
|- ( ta -> et )
syl3an.4
|- ( ( ps /\ th /\ et ) -> ze )
Assertion syl3an
|- ( ( ph /\ ch /\ ta ) -> ze )

Proof

Step Hyp Ref Expression
1 syl3an.1
 |-  ( ph -> ps )
2 syl3an.2
 |-  ( ch -> th )
3 syl3an.3
 |-  ( ta -> et )
4 syl3an.4
 |-  ( ( ps /\ th /\ et ) -> ze )
5 1 2 3 3anim123i
 |-  ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) )
6 5 4 syl
 |-  ( ( ph /\ ch /\ ta ) -> ze )