Theorem eqsstrd 3215

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 3208	. 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 3153

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 3159  df-ss 3166

This theorem is referenced by:  eqsstrrd  3216  eqsstrdi  3231  tfisi  4619  funresdfunsnss  5761  suppssof1  6148  pw2f1odclem  6890  phplem4dom  6918  fival  7029  fiuni  7037  cardonle  7247  exmidfodomrlemim  7261  frecuzrdgtclt  10492  4sqlem19  12547  ennnfonelemkh  12569  ennnfonelemf1  12575  strfvssn  12640  setscom  12658  imasaddfnlemg  12897  imasaddflemg  12899  reldvdsrsrg  13588  znleval  14141  tgrest  14337  resttopon  14339  rest0  14347  lmtopcnp  14418  metequiv2  14664  xmettx  14678  ellimc3apf  14814  dvfvalap  14835  dvcjbr  14857  dvcj  14858  dvfre  14859  nnsf  15495