Theorem sseq2d 3054

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
sseq2d	|- ( ph -> ( C C_ A <-> C C_ B ) )

Proof of Theorem sseq2d

Step	Hyp	Ref	Expression
1		sseq1d.1	. 2 |- ( ph -> A = B )
2		sseq2 3048	. 2 |- ( A = B -> ( C C_ A <-> C C_ B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C C_ A <-> C C_ B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   C_ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  sseq12d  3055  sseqtrd  3062  onsucsssucexmid  4343  sbcrel  4524  funimass2  5092  fnco  5122  fnssresb  5126  fnimaeq0  5135  foimacnv  5271  fvelimab  5360  ssimaexg  5366  fvmptss2  5379  rdgss  6148  isummolem2  10772  isummo  10773  zisum  10774  fsum3cvg3  10789