Theorem sseqtrri 3259

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 3258	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1395   ⊆ wss 3197

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 3203  df-ss 3210

This theorem is referenced by:  eqimss2i  3281  difdif2ss  3461  snsspr1  3816  snsspr2  3817  snsstp1  3818  snsstp2  3819  snsstp3  3820  prsstp12  3821  prsstp13  3822  prsstp23  3823  iunxdif2  4014  pwpwssunieq  4054  sssucid  4506  opabssxp  4793  dmresi  5060  cnvimass  5091  ssrnres  5171  cnvcnv  5181  cnvssrndm  5250  dmmpossx  6351  tfrcllemssrecs  6504  sucinc  6599  mapex  6809  exmidpw  7078  exmidpweq  7079  casefun  7260  djufun  7279  pw1ne1  7422  ressxr  8198  ltrelxr  8215  nnssnn0  9380  un0addcl  9410  un0mulcl  9411  nn0ssxnn0  9443  fzssnn  10272  fzossnn0  10381  isumclim3  11942  isprm3  12648  phimullem  12755  tgvalex  13304  eqgfval  13767  cnfldbas  14532  mpocnfldadd  14533  mpocnfldmul  14535  cnfldcj  14537  cnfldtset  14538  cnfldle  14539  cnfldds  14540  cnrest2  14918  qtopbasss  15203  tgqioo  15237