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Theorem uneq12i 3749
Description: Equality inference for the 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 3746 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 707 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cun 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565
This theorem is referenced by:  indir  3857  difundir  3862  difindi  3863  dfsymdif3  3875  unrab  3880  rabun2  3888  elnelun  3944  dfif6  4067  dfif3  4078  dfif5  4080  symdif0  4570  symdifid  4572  unopab  4700  xpundi  5142  xpundir  5143  xpun  5147  dmun  5301  resundi  5379  resundir  5380  cnvun  5507  rnun  5510  imaundi  5514  imaundir  5515  dmtpop  5580  coundi  5605  coundir  5606  unidmrn  5634  dfdm2  5636  predun  5673  mptun  5992  resasplit  6041  fresaun  6042  fresaunres2  6043  residpr  6374  fpr  6386  fvsnun2  6414  sbthlem5  8034  1sdom  8123  cdaassen  8964  fz0to3un2pr  12398  fz0to4untppr  12399  fzo0to42pr  12512  hashgval  13076  hashinf  13078  relexpcnv  13725  bpoly3  14733  vdwlem6  15633  setsres  15841  xpsc  16157  lefld  17166  opsrtoslem1  19424  volun  23253  iblcnlem1  23494  ex-dif  27168  ex-in  27170  ex-pw  27174  ex-xp  27181  ex-cnv  27182  ex-rn  27185  partfun  29359  ordtprsuni  29789  indval2  29900  sigaclfu2  30007  eulerpartgbij  30257  subfacp1lem1  30922  subfacp1lem5  30927  fixun  31711  refssfne  32048  onint1  32143  bj-pr1un  32691  bj-pr21val  32701  bj-pr2un  32705  bj-pr22val  32707  poimirlem16  33096  poimirlem19  33099  itg2addnclem2  33133  iblabsnclem  33144  rclexi  37442  rtrclex  37444  cnvrcl0  37452  dfrtrcl5  37456  dfrcl2  37486  dfrcl4  37488  iunrelexp0  37514  relexpiidm  37516  corclrcl  37519  relexp01min  37525  corcltrcl  37551  cotrclrcl  37554  frege131d  37576  df3o3  37844  rnfdmpr  40627  31prm  40841
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