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Theorem eleqtri 2154
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 2146 . 2 (𝐴𝐵𝐴𝐶)
41, 3mpbi 143 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by:  eleqtrri  2155  3eltr3i  2160  prid2  3507  2eluzge0  8744  fz01or  9204
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