Theorem eqsstrrd 3277

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 3276	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3213

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 3219  df-ss 3226

This theorem is referenced by:  ssxpbm  5200  ssxp1  5201  ssxp2  5202  suppssof1  6286  tfrlemiubacc  6563  tfr1onlemubacc  6579  tfrcllemubacc  6592  oaword1  6706  phplem4dom  7118  fisseneq  7197  nnnninfeq2  7422  archnqq  7734  hashdmprop2dom  11220  imasaddfnlemg  13544  resmhm2  13718  ringidss  14190  subrg1  14393  subrgdvds  14397  subrguss  14398  subrginv  14399  islss3  14544  lspsnneg  14585  epttop  14972  metequiv2  15378  limccnpcntop  15557  limccnp2lem  15558  limccnp2cntop  15559  umgredgprv  16127  uspgrupgrushgr  16194  usgrumgruspgr  16197  nnsf  16800