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Theorem uneq12i 3150
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 3147 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 417 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001
This theorem is referenced by:  indir  3246  difundir  3250  symdif1  3262  unrab  3268  rabun2  3276  dfif6  3391  dfif3  3402  unopab  3909  xpundi  4482  xpundir  4483  xpun  4487  dmun  4631  resundi  4714  resundir  4715  cnvun  4824  rnun  4827  imaundi  4831  imaundir  4832  dmtpop  4893  coundi  4919  coundir  4920  unidmrn  4950  dfdm2  4952  mptun  5130  fpr  5463  fvsnun2  5479  sbthlemi5  6649  djuunr  6737  casedm  6756  djudm  6764  fzo0to42pr  9596
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