ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eleqtri GIF version

Theorem eleqtri 2264
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 2256 . 2 (𝐴𝐵𝐴𝐶)
41, 3mpbi 145 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185
This theorem is referenced by:  eleqtrri  2265  3eltr3i  2270  prid2  3714  2eluzge0  9605  fz01or  10141  fz0to4untppr  10154  ef0lem  11700  ege2le3  11711  efgt1p2  11735  efgt1p  11736  phi1  12251  cnrehmeocntop  14553  dvcjbr  14632  fmelpw1o  15019
  Copyright terms: Public domain W3C validator