Theorem eqsstrrd 3217

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 2199	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3216	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3154

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167

This theorem is referenced by:  ssxpbm  5102  ssxp1  5103  ssxp2  5104  suppssof1  6150  tfrlemiubacc  6385  tfr1onlemubacc  6401  tfrcllemubacc  6414  oaword1  6526  phplem4dom  6920  fisseneq  6990  nnnninfeq2  7190  archnqq  7479  imasaddfnlemg  12900  resmhm2  13063  ringidss  13528  subrg1  13730  subrgdvds  13734  subrguss  13735  subrginv  13736  islss3  13878  lspsnneg  13919  epttop  14269  metequiv2  14675  limccnpcntop  14854  limccnp2lem  14855  limccnp2cntop  14856  nnsf  15565