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Theorem sneqd 3388
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 3386 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2syl 14 1  |-  ( ph  ->  { A }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-sn 3381
This theorem is referenced by:  dmsnsnsng  4798  cnvsng  4806  ressn  4858  f1osng  5167  fsng  5336  fnressn  5349  fvsng  5359  2nd1st  5806  dfmpt2  5844  cnvf1olem  5845  tpostpos  5879  tfrlemi1  5946  en1bg  6280  xpassen  6304  fztp  8938  fzsuc2  8939  fseq1p1m1  8954  fseq1m1p1  8955
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