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Theorem sneqd 3444
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 3442 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2syl 14 1  |-  ( ph  ->  { A }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   {csn 3431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-sn 3437
This theorem is referenced by:  dmsnsnsng  4874  cnvsng  4882  ressn  4937  f1osng  5257  fsng  5433  fnressn  5446  fvsng  5456  2nd1st  5907  dfmpt2  5945  cnvf1olem  5946  tpostpos  5983  tfrlemi1  6051  tfr1onlemaccex  6067  tfrcllemaccex  6080  en1bg  6469  mapsnen  6480  xpassen  6498  fztp  9422  fzsuc2  9423  fseq1p1m1  9438  fseq1m1p1  9439  zfz1isolemsplit  10139  zfz1isolem1  10141  divalgmod  10802
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