Theorem eqsstrrd 3264

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
eqsstrrd.1	|- ( ph -> B = A )
eqsstrrd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
eqsstrrd	|- ( ph -> A C_ C )

Proof of Theorem eqsstrrd

Step	Hyp	Ref	Expression
1		eqsstrrd.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2237	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3263	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1397   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  ssxpbm  5172  ssxp1  5173  ssxp2  5174  suppssof1  6253  tfrlemiubacc  6496  tfr1onlemubacc  6512  tfrcllemubacc  6525  oaword1  6639  phplem4dom  7048  fisseneq  7127  nnnninfeq2  7328  archnqq  7637  hashdmprop2dom  11109  imasaddfnlemg  13415  resmhm2  13589  ringidss  14061  subrg1  14264  subrgdvds  14268  subrguss  14269  subrginv  14270  islss3  14412  lspsnneg  14453  epttop  14833  metequiv2  15239  limccnpcntop  15418  limccnp2lem  15419  limccnp2cntop  15420  umgredgprv  15985  uspgrupgrushgr  16052  usgrumgruspgr  16055  nnsf  16658