Theorem sseqtrid

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		sseqtrid.1	|- B C_ A
2		sseqtrid.2	|- ( ph -> A = C )
3		sseq2	|- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa	|- ( ( A = C /\ B C_ A ) -> B C_ C )
5	2 1 4	sylancl	|- ( ph -> B C_ C )