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Theorem sneqd 3604
Description: Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
Hypothesis
Ref Expression
sneqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
sneqd  |-  ( ph  ->  { A }  =  { B } )

Proof of Theorem sneqd
StepHypRef Expression
1 sneqd.1 . 2  |-  ( ph  ->  A  =  B )
2 sneq 3602 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2syl 14 1  |-  ( ph  ->  { A }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   {csn 3591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-sn 3597
This theorem is referenced by:  dmsnsnsng  5101  cnvsng  5109  ressn  5164  f1osng  5497  fsng  5684  fnressn  5697  fvsng  5707  2nd1st  6174  dfmpo  6217  cnvf1olem  6218  tpostpos  6258  tfrlemi1  6326  tfr1onlemaccex  6342  tfrcllemaccex  6355  elixpsn  6728  ixpsnf1o  6729  en1bg  6793  mapsnen  6804  xpassen  6823  fztp  10051  fzsuc2  10052  fseq1p1m1  10067  fseq1m1p1  10068  zfz1isolemsplit  10789  zfz1isolem1  10791  fsumm1  11395  fprodm1  11577  divalgmod  11902  ennnfonelemg  12374  ennnfonelemp1  12377  ennnfonelem1  12378  ennnfonelemnn0  12393  setsvalg  12462  strsetsid  12465  mulgval  12862  isunitd  13087  txdis  13410
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