Theorem sseqtrri 3260

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Hypotheses

Ref	Expression
sseqtrri.1	⊢ 𝐴 ⊆ 𝐵
sseqtrri.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
sseqtrri	⊢ 𝐴 ⊆ 𝐶

Proof of Theorem sseqtrri

Step	Hyp	Ref	Expression
1		sseqtrri.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		sseqtrri.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2233	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtri 3259	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1395   ⊆ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  eqimss2i  3282  difdif2ss  3462  snsspr1  3819  snsspr2  3820  snsstp1  3821  snsstp2  3822  snsstp3  3823  prsstp12  3824  prsstp13  3825  prsstp23  3826  iunxdif2  4017  pwpwssunieq  4057  sssucid  4510  opabssxp  4798  dmresi  5066  cnvimass  5097  ssrnres  5177  cnvcnv  5187  cnvssrndm  5256  dmmpossx  6359  tfrcllemssrecs  6513  sucinc  6608  mapex  6818  exmidpw  7095  exmidpweq  7096  casefun  7278  djufun  7297  pw1ne1  7440  ressxr  8216  ltrelxr  8233  nnssnn0  9398  un0addcl  9428  un0mulcl  9429  nn0ssxnn0  9461  fzssnn  10296  fzossnn0  10405  isumclim3  11977  isprm3  12683  phimullem  12790  tgvalex  13339  eqgfval  13802  cnfldbas  14567  mpocnfldadd  14568  mpocnfldmul  14570  cnfldcj  14572  cnfldtset  14573  cnfldle  14574  cnfldds  14575  cnrest2  14953  qtopbasss  15238  tgqioo  15272