Theorem eqsstrrd 3220

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 2202	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3219	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  ssxpbm  5105  ssxp1  5106  ssxp2  5107  suppssof1  6153  tfrlemiubacc  6388  tfr1onlemubacc  6404  tfrcllemubacc  6417  oaword1  6529  phplem4dom  6923  fisseneq  6995  nnnninfeq2  7195  archnqq  7484  imasaddfnlemg  12957  resmhm2  13120  ringidss  13585  subrg1  13787  subrgdvds  13791  subrguss  13792  subrginv  13793  islss3  13935  lspsnneg  13976  epttop  14326  metequiv2  14732  limccnpcntop  14911  limccnp2lem  14912  limccnp2cntop  14913  nnsf  15649