Metamath Proof Explorer

Theorem sseq12

Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999)

Ref Expression
Assertion sseq12 
|- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

Proof

Step Hyp Ref Expression
1 sseq1 
 |-  ( A = B -> ( A C_ C <-> B C_ C ) )
2 sseq2 
 |-  ( C = D -> ( B C_ C <-> B C_ D ) )
3 1 2 sylan9bb 
 |-  ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )