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  4507  opabssxp  4795  dmresi  5063  cnvimass  5094  ssrnres  5174  cnvcnv  5184  cnvssrndm  5253  dmmpossx  6356  tfrcllemssrecs  6509  sucinc  6604  mapex  6814  exmidpw  7086  exmidpweq  7087  casefun  7268  djufun  7287  pw1ne1  7430  ressxr  8206  ltrelxr  8223  nnssnn0  9388  un0addcl  9418  un0mulcl  9419  nn0ssxnn0  9451  fzssnn  10281  fzossnn0  10390  isumclim3  11955  isprm3  12661  phimullem  12768  tgvalex  13317  eqgfval  13780  cnfldbas  14545  mpocnfldadd  14546  mpocnfldmul  14548  cnfldcj  14550  cnfldtset  14551  cnfldle  14552  cnfldds  14553  cnrest2  14931  qtopbasss  15216  tgqioo  15250