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

Theorem eleqtrri 2129
Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eleqtrr.1 𝐴𝐵
eleqtrr.2 𝐶 = 𝐵
Assertion
Ref Expression
eleqtrri 𝐴𝐶

Proof of Theorem eleqtrri
StepHypRef Expression
1 eleqtrr.1 . 2 𝐴𝐵
2 eleqtrr.2 . . 3 𝐶 = 𝐵
32eqcomi 2060 . 2 𝐵 = 𝐶
41, 3eleqtri 2128 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  3eltr4i  2135  opi1  3997  opi2  3998  ordpwsucexmid  4322  peano1  4345  acexmidlemcase  5535  acexmidlem2  5537  ac6sfi  6383  1lt2pi  6496  prarloclemarch2  6575  prarloclemlt  6649  prarloclemcalc  6658  pnfxr  8793  mnfxr  8795
  Copyright terms: Public domain W3C validator