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Metamath Proof Explorer

Theorem eqsstrrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

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

Proof

Step Hyp Ref Expression
1 eqsstrrd.1 
 |-  ( ph -> B = A )
2 eqsstrrd.2 
 |-  ( ph -> B C_ C )
3 1 eqcomd 
 |-  ( ph -> A = B )
4 3 2 eqsstrd 
 |-  ( ph -> A C_ C )