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Theorem uneq12d 3201
Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
uneq1d.1  |-  ( ph  ->  A  =  B )
uneq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
uneq12d  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  D ) )

Proof of Theorem uneq12d
StepHypRef Expression
1 uneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 uneq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 uneq12 3195 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
41, 2, 3syl2anc 408 1  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    u. cun 3039
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045
This theorem is referenced by:  disjpr2  3557  diftpsn3  3631  iunxprg  3863  undifexmid  4087  exmidundif  4099  exmidundifim  4100  suceq  4294  rnpropg  4988  fntpg  5149  foun  5354  fnimapr  5449  fprg  5571  fsnunfv  5589  fsnunres  5590  tfrlemi1  6197  tfr1onlemaccex  6213  tfrcllemaccex  6226  ereq1  6404  undifdc  6780  unfiin  6782  djueq12  6892  fztp  9826  fzsuc2  9827  fseq1p1m1  9842  ennnfonelemg  11843  ennnfonelemp1  11846  ennnfonelem1  11847  ennnfonelemnn0  11862  setsvalg  11916  setsfun0  11922  setsresg  11924  setsslid  11936  exmid1stab  13122
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