ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqeq12 GIF version

Theorem eqeq12 2183
Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
eqeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeq12
StepHypRef Expression
1 eqeq1 2177 . 2 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
2 eqeq2 2180 . 2 (𝐶 = 𝐷 → (𝐵 = 𝐶𝐵 = 𝐷))
31, 2sylan9bb 459 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163
This theorem is referenced by:  eqeq12i  2184  eqeq12d  2185  eqeqan12d  2186  funopg  5230  tfri3  6344  th3qlem1  6613  xpdom2  6807  difinfsnlem  7074  difinfsn  7075  xrlttri3  9747  bcn1  10685  summodc  11339  prodmodc  11534
  Copyright terms: Public domain W3C validator