Theorem eqsstrd 3133

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 3126	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  eqsstrrd  3134  eqsstrdi  3149  tfisi  4501  funresdfunsnss  5623  suppssof1  5999  phplem4dom  6756  fival  6858  fiuni  6866  cardonle  7043  exmidfodomrlemim  7057  frecuzrdgtclt  10194  ennnfonelemkh  11925  ennnfonelemf1  11931  strfvssn  11981  setscom  11999  tgrest  12338  resttopon  12340  rest0  12348  lmtopcnp  12419  metequiv2  12665  xmettx  12679  ellimc3apf  12798  dvfvalap  12819  dvcjbr  12841  dvcj  12842  dvfre  12843  nnsf  13199