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Theorem uneq12i 3416
 Description: Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
uneq1i.1 A = B
uneq12i.2 C = D
Assertion
Ref Expression
uneq12i (AC) = (BD)

Proof of Theorem uneq12i
StepHypRef Expression
1 uneq1i.1 . 2 A = B
2 uneq12i.2 . 2 C = D
3 uneq12 3413 . 2 ((A = B C = D) → (AC) = (BD))
41, 2, 3mp2an 653 1 (AC) = (BD)
 Colors of variables: wff setvar class Syntax hints:   = 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:  indir  3503  difundir  3508  difindi  3509  symdif1  3519  unrab  3526  rabun2  3534  dfun4  3546  iunin  3547  dfif6  3665  dfif3  3672  dfif5  3674  nnc0suc  4412  nnsucelrlem3  4426  ltfintrilem1  4465  dfop2  4575  phiun  4614  opeq  4619  unopab  4638  xpundi  4832  xpundir  4833  xpun  4834  resundi  4981  resundir  4982  cnvun  5033  rnun  5036  imaundi  5039  imaundir  5040  dmtpop  5071  coundi  5082  coundir  5083  fpr  5437  fvsnun2  5448  clos1basesuc  5882  ce2  6192  sbthlem1  6203
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