Metamath Proof Explorer

Theorem 3sstr4i

Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996) (Proof shortened by Eric Schmidt, 26-Jan-2007)

Ref Expression
Hypotheses 3sstr4.1 
|- A C_ B
3sstr4.2 
|- C = A
3sstr4.3 
|- D = B
Assertion 3sstr4i 
|- C C_ D

Proof

Step Hyp Ref Expression
1 3sstr4.1 
 |-  A C_ B
2 3sstr4.2 
 |-  C = A
3 3sstr4.3 
 |-  D = B
4 2 3 sseq12i 
 |-  ( C C_ D <-> A C_ B )
5 1 4 mpbir 
 |-  C C_ D