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  4635  funresdfunsnss  5787  suppssof1  6176  pw2f1odclem  6931  phplem4dom  6959  fival  7072  fiuni  7080  cardonle  7294  exmidfodomrlemim  7309  frecuzrdgtclt  10566  4sqlem19  12732  ennnfonelemkh  12783  ennnfonelemf1  12789  strfvssn  12854  setscom  12872  imasaddfnlemg  13146  imasaddflemg  13148  reldvdsrsrg  13854  znleval  14415  tgrest  14641  resttopon  14643  rest0  14651  lmtopcnp  14722  metequiv2  14968  xmettx  14982  ellimc3apf  15132  dvfvalap  15153  dvcjbr  15180  dvcj  15181  dvfre  15182  nnsf  15942