 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  djueq2 Structured version   Visualization version   GIF version

Theorem djueq2 9186
 Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
djueq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem djueq2
StepHypRef Expression
1 eqid 2795 . 2 𝐶 = 𝐶
2 djueq12 9184 . 2 ((𝐶 = 𝐶𝐴 = 𝐵) → (𝐶𝐴) = (𝐶𝐵))
31, 2mpan 686 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1522   ⊔ cdju 9178 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-un 3868  df-opab 5029  df-xp 5454  df-dju 9181 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator