Theorem sseq1i 3183

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 3180	. 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 1353   C_ wss 3131

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  eqsstri  3189  eqsstrid  3203  ssab  3227  rabss  3234  uniiunlem  3246  prss  3750  prssg  3751  tpss  3760  iunss  3929  pwtr  4221  ordsucss  4505  elomssom  4606  cores2  5143  dffun2  5228  funimaexglem  5301  idref  5759  ordgt0ge1  6438  3nsssucpw1  7237  prarloclemn  7500  bdeqsuc  14672  bj-omssind  14726