Theorem sseq1i 3274

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 3271	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
3	1, 2	ax-mp 5	1 |- ( A C_ C <-> B C_ C )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  eqsstri  3280  eqsstrid  3294  ssab  3318  rabss  3325  uniiunlem  3338  prss  3871  prssg  3872  tpss  3883  iunss  4053  pwtr  4359  ordsucss  4651  elomssom  4752  cores2  5300  dffun2  5387  funimaexglem  5464  idref  5962  ordgt0ge1  6708  3nsssucpw1  7595  prarloclemn  7866  ausgrusgrben  16409  bdeqsuc  16907  bj-omssind  16961