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Theorem sneqd 3419
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 3417 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2syl 14 1  |-  ( ph  ->  { A }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   {csn 3406
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-sn 3412
This theorem is referenced by:  dmsnsnsng  4828  cnvsng  4836  ressn  4888  f1osng  5198  fsng  5368  fnressn  5381  fvsng  5391  2nd1st  5837  dfmpt2  5875  cnvf1olem  5876  tpostpos  5913  tfrlemi1  5981  tfr1onlemaccex  5997  tfrcllemaccex  6010  en1bg  6347  xpassen  6374  fztp  9171  fzsuc2  9172  fseq1p1m1  9187  fseq1m1p1  9188  divalgmod  10471
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