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  5846  suppssof1  6242  pw2f1odclem  7003  phplem4dom  7031  fival  7148  fiuni  7156  cardonle  7370  exmidfodomrlemim  7390  frecuzrdgtclt  10655  4sqlem19  12948  ennnfonelemkh  12999  ennnfonelemf1  13005  strfvssn  13070  setscom  13088  imasaddfnlemg  13363  imasaddflemg  13365  znleval  14633  tgrest  14859  resttopon  14861  rest0  14869  lmtopcnp  14940  metequiv2  15186  xmettx  15200  ellimc3apf  15350  dvfvalap  15371  dvcjbr  15398  dvcj  15399  dvfre  15400  uhgredgm  15950  upgredgssen  15953  umgredgssen  15954  edgumgren  15956  usgredgssen  15976  nnsf  16459