Metamath Proof Explorer


Theorem syl3an2

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995) (Proof shortened by Wolf Lammen, 26-Jun-2022)

Ref Expression
Hypotheses syl3an2.1
|- ( ph -> ch )
syl3an2.2
|- ( ( ps /\ ch /\ th ) -> ta )
Assertion syl3an2
|- ( ( ps /\ ph /\ th ) -> ta )

Proof

Step Hyp Ref Expression
1 syl3an2.1
 |-  ( ph -> ch )
2 syl3an2.2
 |-  ( ( ps /\ ch /\ th ) -> ta )
3 1 3anim2i
 |-  ( ( ps /\ ph /\ th ) -> ( ps /\ ch /\ th ) )
4 3 2 syl
 |-  ( ( ps /\ ph /\ th ) -> ta )