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Theorem uneq12i 3089
 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 3086 . 2 ((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
41, 2, 3mp2an 402 1 (A𝐶) = (B𝐷)
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∪ cun 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916 This theorem is referenced by:  indir  3180  difundir  3184  symdif1  3196  unrab  3202  rabun2  3210  dfif6  3327  dfif3  3337  unopab  3827  xpundi  4339  xpundir  4340  xpun  4344  dmun  4485  resundi  4568  resundir  4569  cnvun  4672  rnun  4675  imaundi  4679  imaundir  4680  dmtpop  4739  coundi  4765  coundir  4766  unidmrn  4793  dfdm2  4795  mptun  4972  fpr  5288  fvsnun2  5304  fzo0to42pr  8826
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