Theorem eqsstrd 3237

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 3230	. 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 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  eqsstrrd  3238  eqsstrdi  3253  tfisi  4653  funresdfunsnss  5810  suppssof1  6199  pw2f1odclem  6956  phplem4dom  6984  fival  7098  fiuni  7106  cardonle  7320  exmidfodomrlemim  7340  frecuzrdgtclt  10603  4sqlem19  12847  ennnfonelemkh  12898  ennnfonelemf1  12904  strfvssn  12969  setscom  12987  imasaddfnlemg  13261  imasaddflemg  13263  reldvdsrsrg  13969  znleval  14530  tgrest  14756  resttopon  14758  rest0  14766  lmtopcnp  14837  metequiv2  15083  xmettx  15097  ellimc3apf  15247  dvfvalap  15268  dvcjbr  15295  dvcj  15296  dvfre  15297  uhgredgm  15842  edgumgren  15846  nnsf  16144