Theorem eqsstrd 3216

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 3209	. 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 3154

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 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167

This theorem is referenced by:  eqsstrrd  3217  eqsstrdi  3232  tfisi  4620  funresdfunsnss  5762  suppssof1  6150  pw2f1odclem  6892  phplem4dom  6920  fival  7031  fiuni  7039  cardonle  7249  exmidfodomrlemim  7263  frecuzrdgtclt  10495  4sqlem19  12550  ennnfonelemkh  12572  ennnfonelemf1  12578  strfvssn  12643  setscom  12661  imasaddfnlemg  12900  imasaddflemg  12902  reldvdsrsrg  13591  znleval  14152  tgrest  14348  resttopon  14350  rest0  14358  lmtopcnp  14429  metequiv2  14675  xmettx  14689  ellimc3apf  14839  dvfvalap  14860  dvcjbr  14887  dvcj  14888  dvfre  14889  nnsf  15565