MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsn Structured version   Visualization version   GIF version

Theorem nfsn 4635
Description: Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
Hypothesis
Ref Expression
nfsn.1 𝑥𝐴
Assertion
Ref Expression
nfsn 𝑥{𝐴}

Proof of Theorem nfsn
StepHypRef Expression
1 dfsn2 4570 . 2 {𝐴} = {𝐴, 𝐴}
2 nfsn.1 . . 3 𝑥𝐴
32, 2nfpr 4620 . 2 𝑥{𝐴, 𝐴}
41, 3nfcxfr 2972 1 𝑥{𝐴}
Colors of variables: wff setvar class
Syntax hints:  wnfc 2958  {csn 4557  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  nfop  4811  iunopeqop  5402  nfpred  6146  nfsuc  6255  sniota  6339  dfmpo  7786  bnj958  32111  bnj1000  32112  bnj1446  32214  bnj1447  32215  bnj1448  32216  bnj1466  32222  bnj1467  32223  nosupbnd2  33113  nfaltop  33338  stoweidlem21  42183  stoweidlem47  42209  nfdfat  43203
  Copyright terms: Public domain W3C validator