Theorem eleqtri 2829

Description: Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.)

Hypotheses

Ref	Expression
eleqtri.1	⊢ 𝐴 ∈ 𝐵
eleqtri.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
eleqtri	⊢ 𝐴 ∈ 𝐶

Proof of Theorem eleqtri

Step	Hyp	Ref	Expression
1		eleqtri.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		eleqtri.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	eleq2i 2823	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 230	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1541   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806

This theorem is referenced by:  eleqtrri  2830  3eltr3i  2843  prid2  4711  2eluzge0  12774  faclbnd4lem1  14195  cats1fv  14761  bpoly2  15959  bpoly3  15960  bpoly4  15961  ef0lem  15980  phi1  16679  gsumws1  18741  lt6abl  19802  uvcvvcl  21719  mhpvarcl  22058  smadiadetlem4  22579  indiscld  23001  cnrehmeo  24873  cnrehmeoOLD  24874  ovolicc1  25439  dvcjbr  25875  vieta1lem2  26241  dvloglem  26579  logdmopn  26580  efopnlem2  26588  cxpcn  26676  cxpcnOLD  26677  loglesqrt  26693  log2ublem2  26879  efrlim  26901  efrlimOLD  26902  precsexlem11  28150  tgcgr4  28504  axlowdimlem16  28930  axlowdimlem17  28931  nlelchi  32033  hmopidmchi  32123  indf  32828  evl1deg2  33532  evl1deg3  33533  raddcn  33934  xrge0tmd  33950  ballotlem1ri  34540  chtvalz  34634  circlemethhgt  34648  dvtanlem  37709  ftc1cnnc  37732  dvasin  37744  dvacos  37745  dvreasin  37746  dvreacos  37747  areacirclem2  37749  areacirclem4  37751  cncfres  37805  resuppsinopn  42396  jm2.23  43029  0finon  43481  1finon  43482  2finon  43483  3finon  43484  4finon  43485  fvnonrel  43630  frege54cor1c  43948  fourierdlem28  46173  fourierdlem57  46201  fourierdlem59  46203  fourierdlem62  46206  fourierdlem68  46212  fouriersw  46269  etransclem23  46295  etransclem35  46307  etransclem38  46310  etransclem39  46311  etransclem44  46316  etransclem45  46317  etransclem47  46319  rrxtopn0  46331  hoidmvlelem2  46634  vonicclem2  46722  fmtno4prmfac  47603  gpg5grlim  48124  gpg5grlic  48125