Theorem eqsstrrd 3279

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 2240	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3278	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3214

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  ssxpbm  5203  ssxp1  5204  ssxp2  5205  suppssof1  6293  tfrlemiubacc  6574  tfr1onlemubacc  6590  tfrcllemubacc  6603  oaword1  6717  phplem4dom  7129  fisseneq  7208  nnnninfeq2  7433  archnqq  7748  hashdmprop2dom  11241  imasaddfnlemg  13578  resmhm2  13743  ringidss  14272  subrg1  14477  subrgdvds  14481  subrguss  14482  subrginv  14483  islss3  14653  lspsnneg  14694  epttop  15081  metequiv2  15487  limccnpcntop  15666  limccnp2lem  15667  limccnp2cntop  15668  umgredgprv  16236  uspgrupgrushgr  16303  usgrumgruspgr  16306  nnsf  16909