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Theorem eleqtri 2112
Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eleqtr.1 𝐴𝐵
eleqtr.2 𝐵 = 𝐶
Assertion
Ref Expression
eleqtri 𝐴𝐶

Proof of Theorem eleqtri
StepHypRef Expression
1 eleqtr.1 . 2 𝐴𝐵
2 eleqtr.2 . . 3 𝐵 = 𝐶
32eleq2i 2104 . 2 (𝐴𝐵𝐴𝐶)
41, 3mpbi 133 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = 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:  eleqtrri  2113  3eltr3i  2118  prid2  3477  2eluzge0  8517
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