Theorem sseq1d 3056

Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
sseq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
sseq1d	|- ( ph -> ( A C_ C <-> B C_ C ) )

Proof of Theorem sseq1d

Step	Hyp	Ref	Expression
1		sseq1d.1	. 2 |- ( ph -> A = B )
2		sseq1 3050	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A C_ C <-> B C_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   C_ wss 3002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3008  df-ss 3015

This theorem is referenced by:  sseq12d  3058  eqsstrd  3063  snssg  3581  ssiun2s  3782  treq  3950  onsucsssucexmid  4358  funimass1  5106  feq1  5160  sbcfg  5175  fvmptssdm  5402  fvimacnvi  5429  nnsucsssuc  6269  ereq1  6315  elpm2r  6439  nnnninf  6869