Theorem eqsstrd 3219

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 3212	. 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 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  eqsstrrd  3220  eqsstrdi  3235  tfisi  4623  funresdfunsnss  5765  suppssof1  6153  pw2f1odclem  6895  phplem4dom  6923  fival  7036  fiuni  7044  cardonle  7254  exmidfodomrlemim  7268  frecuzrdgtclt  10513  4sqlem19  12578  ennnfonelemkh  12629  ennnfonelemf1  12635  strfvssn  12700  setscom  12718  imasaddfnlemg  12957  imasaddflemg  12959  reldvdsrsrg  13648  znleval  14209  tgrest  14405  resttopon  14407  rest0  14415  lmtopcnp  14486  metequiv2  14732  xmettx  14746  ellimc3apf  14896  dvfvalap  14917  dvcjbr  14944  dvcj  14945  dvfre  14946  nnsf  15649