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Theorem eleqtri 2169
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 2161 . 2 (𝐴𝐵𝐴𝐶)
41, 3mpbi 144 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1296  wcel 1445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-cleq 2088  df-clel 2091
This theorem is referenced by:  eleqtrri  2170  3eltr3i  2175  prid2  3569  2eluzge0  9162  fz01or  9674  ef0lem  11099  ege2le3  11110  efgt1p2  11134  efgt1p  11135  phi1  11622
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