Theorem eqsstrrd 3285

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 2244	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3284	1 |- ( ph -> A C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  ssxpbm  5223  ssxp1  5224  ssxp2  5225  suppssof1  6320  tfrlemiubacc  6601  tfr1onlemubacc  6617  tfrcllemubacc  6630  oaword1  6744  phplem4dom  7163  fisseneq  7242  nnnninfeq2  7469  archnqq  7784  hashdmprop2dom  11296  imasaddfnlemg  13635  resmhm2  13795  cmnsubm  14112  ringidss  14334  subrg1  14539  subrgdvds  14543  subrguss  14544  subrginv  14545  islss3  14716  lspsnneg  14757  epttop  15191  metequiv2  15597  limccnpcntop  15776  limccnp2lem  15777  limccnp2cntop  15778  umgredgprv  16356  uspgrupgrushgr  16423  usgrumgruspgr  16426  nnsf  17048