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Theorem eleqtri 2214
 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 2206 . 2 (𝐴𝐵𝐴𝐶)
41, 3mpbi 144 1 𝐴𝐶
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135 This theorem is referenced by:  eleqtrri  2215  3eltr3i  2220  prid2  3630  2eluzge0  9382  fz01or  9903  ef0lem  11378  ege2le3  11389  efgt1p2  11413  efgt1p  11414  phi1  11906  cnrehmeocntop  12776  dvcjbr  12855
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