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

Theorem eleqtrd 2116
 Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
eleqtrd.1 (𝜑𝐴𝐵)
eleqtrd.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
eleqtrd (𝜑𝐴𝐶)

Proof of Theorem eleqtrd
StepHypRef Expression
1 eleqtrd.1 . 2 (𝜑𝐴𝐵)
2 eleqtrd.2 . . 3 (𝜑𝐵 = 𝐶)
32eleq2d 2107 . 2 (𝜑 → (𝐴𝐵𝐴𝐶))
41, 3mpbid 135 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393 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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by:  eleqtrrd  2117  3eltr3d  2120  syl5eleq  2126  syl6eleq  2130  opth1  3973  0nelop  3985  tfisi  4310  ercl  6117  erth  6150  ecelqsdm  6176  phpm  6327  lincmb01cmp  8871  fzopth  8924  fzoaddel2  9049  fzosubel2  9051  fzocatel  9055  zpnn0elfzo1  9064  fzoend  9078  peano2fzor  9088  monoord2  9236  isermono  9237
 Copyright terms: Public domain W3C validator