Theorem sseq1i 3264

Description: An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sseq1i.1	|- A = B

Assertion

Ref	Expression
sseq1i	|- ( A C_ C <-> B C_ C )

Proof of Theorem sseq1i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 |- A = B
2		sseq1 3261	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
3	1, 2	ax-mp 5	1 |- ( A C_ C <-> B C_ C )

Syntax hints:   <-> wb 105   = wceq 1398   C_ wss 3211

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 3217  df-ss 3224

This theorem is referenced by:  eqsstri  3270  eqsstrid  3284  ssab  3308  rabss  3315  uniiunlem  3328  prss  3850  prssg  3851  tpss  3862  iunss  4032  pwtr  4335  ordsucss  4626  elomssom  4727  cores2  5275  dffun2  5362  funimaexglem  5439  idref  5929  ordgt0ge1  6668  3nsssucpw1  7546  prarloclemn  7814  ausgrusgrben  16163  bdeqsuc  16651  bj-omssind  16705