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Theorem eleqtrrd 2117
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrrd.1 (𝜑𝐴𝐵)
eleqtrrd.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
eleqtrrd (𝜑𝐴𝐶)

Proof of Theorem eleqtrrd
StepHypRef Expression
1 eleqtrrd.1 . 2 (𝜑𝐴𝐵)
2 eleqtrrd.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2045 . 2 (𝜑𝐵 = 𝐶)
41, 3eleqtrd 2116 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:  3eltr4d  2121  tfrexlem  5948  erref  6126  en1uniel  6284  fin0  6342  fin0or  6343  prarloclemarch2  6515  fzopth  8922  fzoss2  9026  fzosubel3  9050  elfzomin  9060  elfzonlteqm1  9064  fzoend  9076  fzofzp1  9081  fzofzp1b  9082  peano2fzor  9086  frecuzrdgcl  9173  frecuzrdg0  9174  frecuzrdgsuc  9175
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