 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  uneq12 GIF version

Theorem uneq12 3413
 Description: Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
uneq12 ((A = B C = D) → (AC) = (BD))

Proof of Theorem uneq12
StepHypRef Expression
1 uneq1 3411 . 2 (A = B → (AC) = (BC))
2 uneq2 3412 . 2 (C = D → (BC) = (BD))
31, 2sylan9eq 2405 1 ((A = B C = D) → (AC) = (BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3207 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by:  uneq12i  3416  uneq12d  3419  un00  3586  pw1equn  4331  pw1eqadj  4332  nnsucelr  4428  dmpropg  5068  fnun  5189  fvun  5378
 Copyright terms: Public domain W3C validator