Theorem sseq1i 3163

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 3160	. 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 104   = wceq 1342   C_ wss 3111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-in 3117  df-ss 3124

This theorem is referenced by:  eqsstri  3169  eqsstrid  3183  ssab  3207  rabss  3214  uniiunlem  3226  prss  3723  prssg  3724  tpss  3732  iunss  3901  pwtr  4191  ordsucss  4475  elomssom  4576  cores2  5110  dffun2  5192  funimaexglem  5265  idref  5719  ordgt0ge1  6394  3nsssucpw1  7183  prarloclemn  7431  bdeqsuc  13598  bj-omssind  13652