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Theorem sneqd 3510
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 3508 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2syl 14 1  |-  ( ph  ->  { A }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   {csn 3497
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-sn 3503
This theorem is referenced by:  dmsnsnsng  4986  cnvsng  4994  ressn  5049  f1osng  5376  fsng  5561  fnressn  5574  fvsng  5584  2nd1st  6046  dfmpo  6088  cnvf1olem  6089  tpostpos  6129  tfrlemi1  6197  tfr1onlemaccex  6213  tfrcllemaccex  6226  elixpsn  6597  ixpsnf1o  6598  en1bg  6662  mapsnen  6673  xpassen  6692  fztp  9826  fzsuc2  9827  fseq1p1m1  9842  fseq1m1p1  9843  zfz1isolemsplit  10549  zfz1isolem1  10551  fsumm1  11153  divalgmod  11551  ennnfonelemg  11843  ennnfonelemp1  11846  ennnfonelem1  11847  ennnfonelemnn0  11862  setsvalg  11916  strsetsid  11919  txdis  12373
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