Metamath Proof Explorer

Theorem sstrd

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004)

Ref Expression
Hypotheses sstrd.1 
|- ( ph -> A C_ B )
sstrd.2 
|- ( ph -> B C_ C )
Assertion sstrd 
|- ( ph -> A C_ C )

Proof

Step Hyp Ref Expression
1 sstrd.1 
 |-  ( ph -> A C_ B )
2 sstrd.2 
 |-  ( ph -> B C_ C )
3 sstr 
 |-  ( ( A C_ B /\ B C_ C ) -> A C_ C )
4 1 2 3 syl2anc 
 |-  ( ph -> A C_ C )