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

Theorem eleqtrri 2110
 Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eleqtrr.1 A B
eleqtrr.2 𝐶 = B
Assertion
Ref Expression
eleqtrri A 𝐶

Proof of Theorem eleqtrri
StepHypRef Expression
1 eleqtrr.1 . 2 A B
2 eleqtrr.2 . . 3 𝐶 = B
32eqcomi 2041 . 2 B = 𝐶
41, 3eleqtri 2109 1 A 𝐶
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033 This theorem is referenced by:  3eltr4i  2116  opi1  3960  opi2  3961  ordpwsucexmid  4246  peano1  4260  acexmidlemcase  5450  acexmidlem2  5452  1lt2pi  6324  prarloclemarch2  6402  prarloclemlt  6475  prarloclemcalc  6484  pnfxr  8442  mnfxr  8444
 Copyright terms: Public domain W3C validator