Theorem sseq1i 3218

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 3215	. 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 1372   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  eqsstri  3224  eqsstrid  3238  ssab  3262  rabss  3269  uniiunlem  3281  prss  3788  prssg  3789  tpss  3798  iunss  3967  pwtr  4262  ordsucss  4551  elomssom  4652  cores2  5194  dffun2  5280  funimaexglem  5356  idref  5824  ordgt0ge1  6520  3nsssucpw1  7347  prarloclemn  7611  bdeqsuc  15750  bj-omssind  15804