Theorem eqsstrd 3173

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 3166	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1342   C_ wss 3111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-in 3117  df-ss 3124

This theorem is referenced by:  eqsstrrd  3174  eqsstrdi  3189  tfisi  4558  funresdfunsnss  5682  suppssof1  6061  phplem4dom  6819  fival  6926  fiuni  6934  cardonle  7134  exmidfodomrlemim  7148  frecuzrdgtclt  10346  ennnfonelemkh  12282  ennnfonelemf1  12288  strfvssn  12353  setscom  12371  tgrest  12710  resttopon  12712  rest0  12720  lmtopcnp  12791  metequiv2  13037  xmettx  13051  ellimc3apf  13170  dvfvalap  13191  dvcjbr  13213  dvcj  13214  dvfre  13215  nnsf  13719