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Theorem 3eltr4d 2315
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4d.1 (𝜑𝐴𝐵)
3eltr4d.2 (𝜑𝐶 = 𝐴)
3eltr4d.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3eltr4d (𝜑𝐶𝐷)

Proof of Theorem 3eltr4d
StepHypRef Expression
1 3eltr4d.2 . 2 (𝜑𝐶 = 𝐴)
2 3eltr4d.1 . . 3 (𝜑𝐴𝐵)
3 3eltr4d.3 . . 3 (𝜑𝐷 = 𝐵)
42, 3eleqtrrd 2311 . 2 (𝜑𝐴𝐷)
51, 4eqeltrd 2308 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1397  wcel 2202
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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213
This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227
This theorem is referenced by:  ovmpodxf  6147  nnaordi  6676  iccf1o  10239  ccatw2s1p1g  11226  nnmindc  12623  ennnfonelemrn  13058  ctiunctlemfo  13078  sgrppropd  13514  mndpropd  13541  issubmnd  13543  imasgrp  13716  mulgnndir  13756  subg0cl  13787  subginvcl  13788  subgcl  13789  rngcl  13976  rngpropd  13987  srgcl  14002  srgidcl  14008  ringidcl  14052  ringpropd  14070  dvdsrd  14127  dvrvald  14167  subrngmcl  14242  subrgmcl  14266  subrgunit  14272  lmodprop2d  14381  lidl0  14522  lidl1  14523  psraddcl  14713  wlkl1loop  16228  wlkres  16249  clwwlknonex2lem1  16307
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