Theorem eqsstrd 3260

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 3253	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  eqsstrrd  3261  eqsstrdi  3276  tfisi  4679  funresdfunsnss  5842  suppssof1  6236  pw2f1odclem  6995  phplem4dom  7023  fival  7137  fiuni  7145  cardonle  7359  exmidfodomrlemim  7379  frecuzrdgtclt  10643  4sqlem19  12932  ennnfonelemkh  12983  ennnfonelemf1  12989  strfvssn  13054  setscom  13072  imasaddfnlemg  13347  imasaddflemg  13349  znleval  14617  tgrest  14843  resttopon  14845  rest0  14853  lmtopcnp  14924  metequiv2  15170  xmettx  15184  ellimc3apf  15334  dvfvalap  15355  dvcjbr  15382  dvcj  15383  dvfre  15384  uhgredgm  15934  upgredgssen  15937  umgredgssen  15938  edgumgren  15940  usgredgssen  15960  nnsf  16371