Theorem sseq12i 3265

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
sseq1i.1	|- A = B
sseq12i.2	|- C = D

Assertion

Ref	Expression
sseq12i	|- ( A C_ C <-> B C_ D )

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 |- A = B
2		sseq12i.2	. 2 |- C = D
3		sseq12 3262	. 2 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )
4	1, 2, 3	mp2an 426	1 |- ( A C_ C <-> B C_ D )

Syntax hints:   <-> wb 105   = wceq 1398   C_ wss 3210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3216  df-ss 3223

This theorem is referenced by:  3sstr3i  3277  3sstr4i  3278  3sstr3g  3279  3sstr4g  3280  ss2rab  3313  issubgr  16239