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Theorem uneq12i 3228
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  |-  ( A  u.  C )  =  ( B  u.  D
)

Proof of Theorem uneq12i
StepHypRef Expression
1 uneq1i.1 . 2  |-  A  =  B
2 uneq12i.2 . 2  |-  C  =  D
3 uneq12 3225 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
41, 2, 3mp2an 422 1  |-  ( A  u.  C )  =  ( B  u.  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    u. cun 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075
This theorem is referenced by:  indir  3325  difundir  3329  symdif1  3341  unrab  3347  rabun2  3355  dfif6  3476  dfif3  3487  unopab  4010  xpundi  4598  xpundir  4599  xpun  4603  dmun  4749  resundi  4835  resundir  4836  cnvun  4947  rnun  4950  imaundi  4954  imaundir  4955  dmtpop  5017  coundi  5043  coundir  5044  unidmrn  5074  dfdm2  5076  mptun  5257  fpr  5605  fvsnun2  5621  sbthlemi5  6852  djuunr  6954  djuun  6955  casedm  6974  djudm  6993  djuassen  7085  fzo0to42pr  10021
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