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Theorem eleqtrrd 2133
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 2061 . 2 (𝜑𝐵 = 𝐶)
41, 3eleqtrd 2132 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  3eltr4d  2137  tfrexlem  5979  erref  6157  en1uniel  6315  fin0  6373  fin0or  6374  prarloclemarch2  6575  fzopth  9026  fzoss2  9130  fzosubel3  9154  elfzomin  9164  elfzonlteqm1  9168  fzoend  9180  fzofzp1  9185  fzofzp1b  9186  peano2fzor  9190  zmodfzo  9297  frecuzrdgcl  9363  frecuzrdg0  9364  frecuzrdgsuc  9365  bcn2  9632
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