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Theorem sneqd 3380
Description: Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
Hypothesis
Ref Expression
sneqd.1 (φA = B)
Assertion
Ref Expression
sneqd (φ → {A} = {B})

Proof of Theorem sneqd
StepHypRef Expression
1 sneqd.1 . 2 (φA = B)
2 sneq 3378 . 2 (A = B → {A} = {B})
31, 2syl 14 1 (φ → {A} = {B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {csn 3367
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-sn 3373
This theorem is referenced by:  dmsnsnsng  4741  cnvsng  4749  ressn  4801  f1osng  5110  fsng  5279  fnressn  5292  fvsng  5302  2nd1st  5748  dfmpt2  5786  cnvf1olem  5787  tpostpos  5820  tfrlemi1  5887  en1bg  6216  xpassen  6240  fztp  8710  fzsuc2  8711  fseq1p1m1  8726  fseq1m1p1  8727
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