MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eleq12 Structured version   Visualization version   GIF version

Theorem eleq12 2815
Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
eleq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Proof of Theorem eleq12
StepHypRef Expression
1 eleq1 2813 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 eleq2 2814 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 508 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-clel 2802
This theorem is referenced by:  rru  3772  trel  5278  epelg  5586  preleqg  9654  preleqALT  9656  oemapval  9722  cantnf  9732  wemapwe  9736  nnsdomel  10029  cldval  23010  isufil  23890  taylthlem2  26394  umgr2v2enb1  29455  issiga  33901  bj-epelg  36723  rdgssun  37033  fvineqsneu  37066  matunitlindf  37267  wepwsolem  42640  aomclem8  42659  grumnud  43897  nelbr  46824
  Copyright terms: Public domain W3C validator