Theorem eqsstrd 3206

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
eqsstrd.1	|- ( ph -> A = B )
eqsstrd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
eqsstrd	|- ( ph -> A C_ C )

Proof of Theorem eqsstrd

Step	Hyp	Ref	Expression
1		eqsstrd.2	. 2 |- ( ph -> B C_ C )
2		eqsstrd.1	. . 3 |- ( ph -> A = B )
3	2	sseq1d 3199	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  eqsstrrd  3207  eqsstrdi  3222  tfisi  4604  funresdfunsnss  5740  suppssof1  6125  pw2f1odclem  6863  phplem4dom  6891  fival  7000  fiuni  7008  cardonle  7217  exmidfodomrlemim  7231  frecuzrdgtclt  10454  4sqlem19  12444  ennnfonelemkh  12466  ennnfonelemf1  12472  strfvssn  12537  setscom  12555  imasaddfnlemg  12794  imasaddflemg  12796  reldvdsrsrg  13459  tgrest  14146  resttopon  14148  rest0  14156  lmtopcnp  14227  metequiv2  14473  xmettx  14487  ellimc3apf  14606  dvfvalap  14627  dvcjbr  14649  dvcj  14650  dvfre  14651  nnsf  15233