Theorem sseqtrri 3177

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 2169	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtri 3176	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1343   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  eqimss2i  3199  difdif2ss  3379  snsspr1  3721  snsspr2  3722  snsstp1  3723  snsstp2  3724  snsstp3  3725  prsstp12  3726  prsstp13  3727  prsstp23  3728  iunxdif2  3914  pwpwssunieq  3954  sssucid  4393  opabssxp  4678  dmresi  4939  cnvimass  4967  ssrnres  5046  cnvcnv  5056  cnvssrndm  5125  dmmpossx  6167  tfrcllemssrecs  6320  sucinc  6413  mapex  6620  exmidpw  6874  exmidpweq  6875  casefun  7050  djufun  7069  pw1ne1  7185  ressxr  7942  ltrelxr  7959  nnssnn0  9117  un0addcl  9147  un0mulcl  9148  nn0ssxnn0  9180  fzssnn  10003  fzossnn0  10110  isumclim3  11364  isprm3  12050  phimullem  12157  tgvalex  12690  cnrest2  12876  qtopbasss  13161  tgqioo  13187