Metamath Proof Explorer


Theorem sseqtrrd

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

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

Proof

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