Theorem eqsstrd 3229

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 3222	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  eqsstrrd  3230  eqsstrdi  3245  tfisi  4636  funresdfunsnss  5789  suppssof1  6178  pw2f1odclem  6933  phplem4dom  6961  fival  7074  fiuni  7082  cardonle  7296  exmidfodomrlemim  7311  frecuzrdgtclt  10568  4sqlem19  12765  ennnfonelemkh  12816  ennnfonelemf1  12822  strfvssn  12887  setscom  12905  imasaddfnlemg  13179  imasaddflemg  13181  reldvdsrsrg  13887  znleval  14448  tgrest  14674  resttopon  14676  rest0  14684  lmtopcnp  14755  metequiv2  15001  xmettx  15015  ellimc3apf  15165  dvfvalap  15186  dvcjbr  15213  dvcj  15214  dvfre  15215  nnsf  15979