Theorem eqsstrrd 3261

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 2235	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3260	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  ssxpbm  5164  ssxp1  5165  ssxp2  5166  suppssof1  6242  tfrlemiubacc  6482  tfr1onlemubacc  6498  tfrcllemubacc  6511  oaword1  6625  phplem4dom  7031  fisseneq  7107  nnnninfeq2  7307  archnqq  7615  hashdmprop2dom  11079  imasaddfnlemg  13363  resmhm2  13537  ringidss  14008  subrg1  14211  subrgdvds  14215  subrguss  14216  subrginv  14217  islss3  14359  lspsnneg  14400  epttop  14780  metequiv2  15186  limccnpcntop  15365  limccnp2lem  15366  limccnp2cntop  15367  umgredgprv  15931  uspgrupgrushgr  15996  usgrumgruspgr  15999  nnsf  16459