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Theorem eleqtrd 2116
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrd.1 (𝜑𝐴𝐵)
eleqtrd.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
eleqtrd (𝜑𝐴𝐶)

Proof of Theorem eleqtrd
StepHypRef Expression
1 eleqtrd.1 . 2 (𝜑𝐴𝐵)
2 eleqtrd.2 . . 3 (𝜑𝐵 = 𝐶)
32eleq2d 2107 . 2 (𝜑 → (𝐴𝐵𝐴𝐶))
41, 3mpbid 135 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by:  eleqtrrd  2117  3eltr3d  2120  syl5eleq  2126  syl6eleq  2130  opth1  3970  0nelop  3982  tfisi  4297  ercl  6104  erth  6137  ecelqsdm  6163  phpm  6314  lincmb01cmp  8838  fzopth  8891  fzoaddel2  9016  fzosubel2  9018  fzocatel  9022  zpnn0elfzo1  9031  fzoend  9045  peano2fzor  9055  monoord2  9114  isermono  9115
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