Theorem eqsstrd 3178

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 3171	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  eqsstrrd  3179  eqsstrdi  3194  tfisi  4564  funresdfunsnss  5688  suppssof1  6067  phplem4dom  6828  fival  6935  fiuni  6943  cardonle  7143  exmidfodomrlemim  7157  frecuzrdgtclt  10356  ennnfonelemkh  12345  ennnfonelemf1  12351  strfvssn  12416  setscom  12434  tgrest  12809  resttopon  12811  rest0  12819  lmtopcnp  12890  metequiv2  13136  xmettx  13150  ellimc3apf  13269  dvfvalap  13290  dvcjbr  13312  dvcj  13313  dvfre  13314  nnsf  13885