Theorem eqsstrrd 3265

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 3264	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  ssxpbm  5179  ssxp1  5180  ssxp2  5181  suppssof1  6262  tfrlemiubacc  6539  tfr1onlemubacc  6555  tfrcllemubacc  6568  oaword1  6682  phplem4dom  7091  fisseneq  7170  nnnninfeq2  7388  archnqq  7697  hashdmprop2dom  11171  imasaddfnlemg  13477  resmhm2  13651  ringidss  14123  subrg1  14326  subrgdvds  14330  subrguss  14331  subrginv  14332  islss3  14475  lspsnneg  14516  epttop  14901  metequiv2  15307  limccnpcntop  15486  limccnp2lem  15487  limccnp2cntop  15488  umgredgprv  16056  uspgrupgrushgr  16123  usgrumgruspgr  16126  nnsf  16731