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  4680  funresdfunsnss  5849  suppssof1  6245  pw2f1odclem  7008  phplem4dom  7036  fival  7153  fiuni  7161  cardonle  7375  exmidfodomrlemim  7395  frecuzrdgtclt  10660  4sqlem19  12953  ennnfonelemkh  13004  ennnfonelemf1  13010  strfvssn  13075  setscom  13093  imasaddfnlemg  13368  imasaddflemg  13370  znleval  14638  tgrest  14864  resttopon  14866  rest0  14874  lmtopcnp  14945  metequiv2  15191  xmettx  15205  ellimc3apf  15355  dvfvalap  15376  dvcjbr  15403  dvcj  15404  dvfre  15405  uhgredgm  15955  upgredgssen  15958  umgredgssen  15959  edgumgren  15961  usgredgssen  15981  nnsf  16485