Theorem eqsstrrd 3238

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 2213	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3237	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  ssxpbm  5137  ssxp1  5138  ssxp2  5139  suppssof1  6199  tfrlemiubacc  6439  tfr1onlemubacc  6455  tfrcllemubacc  6468  oaword1  6580  phplem4dom  6984  fisseneq  7057  nnnninfeq2  7257  archnqq  7565  hashdmprop2dom  11026  imasaddfnlemg  13261  resmhm2  13435  ringidss  13906  subrg1  14108  subrgdvds  14112  subrguss  14113  subrginv  14114  islss3  14256  lspsnneg  14297  epttop  14677  metequiv2  15083  limccnpcntop  15262  limccnp2lem  15263  limccnp2cntop  15264  nnsf  16144