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Theorem uneq12d 3226
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 3220 . 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 1331    u. cun 3064
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 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070
This theorem is referenced by:  disjpr2  3582  diftpsn3  3656  iunxprg  3888  undifexmid  4112  exmidundif  4124  exmidundifim  4125  suceq  4319  rnpropg  5013  fntpg  5174  foun  5379  fnimapr  5474  fprg  5596  fsnunfv  5614  fsnunres  5615  tfrlemi1  6222  tfr1onlemaccex  6238  tfrcllemaccex  6251  ereq1  6429  undifdc  6805  unfiin  6807  djueq12  6917  fztp  9851  fzsuc2  9852  fseq1p1m1  9867  ennnfonelemg  11905  ennnfonelemp1  11908  ennnfonelem1  11909  ennnfonelemnn0  11924  setsvalg  11978  setsfun0  11984  setsresg  11986  setsslid  11998  exmid1stab  13184
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