Metamath Proof Explorer


Theorem sylan9ssr

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004)

Ref Expression
Hypotheses sylan9ssr.1
|- ( ph -> A C_ B )
sylan9ssr.2
|- ( ps -> B C_ C )
Assertion sylan9ssr
|- ( ( ps /\ ph ) -> A C_ C )

Proof

Step Hyp Ref Expression
1 sylan9ssr.1
 |-  ( ph -> A C_ B )
2 sylan9ssr.2
 |-  ( ps -> B C_ C )
3 1 2 sylan9ss
 |-  ( ( ph /\ ps ) -> A C_ C )
4 3 ancoms
 |-  ( ( ps /\ ph ) -> A C_ C )