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  6236  tfrlemiubacc  6476  tfr1onlemubacc  6492  tfrcllemubacc  6505  oaword1  6617  phplem4dom  7023  fisseneq  7096  nnnninfeq2  7296  archnqq  7604  hashdmprop2dom  11066  imasaddfnlemg  13347  resmhm2  13521  ringidss  13992  subrg1  14195  subrgdvds  14199  subrguss  14200  subrginv  14201  islss3  14343  lspsnneg  14384  epttop  14764  metequiv2  15170  limccnpcntop  15349  limccnp2lem  15350  limccnp2cntop  15351  umgredgprv  15915  uspgrupgrushgr  15980  usgrumgruspgr  15983  nnsf  16371