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

Theorem nfsn 4189
 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 4138 . 2 {𝐴} = {𝐴, 𝐴}
2 nfsn.1 . . 3 𝑥𝐴
32, 2nfpr 4179 . 2 𝑥{𝐴, 𝐴}
41, 3nfcxfr 2749 1 𝑥{𝐴}
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2738  {csn 4125  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  nfop  4356  iunopeqop  4906  nfpred  5602  nfsuc  5713  sniota  5795  dfmpt2  7154  bnj958  30264  bnj1000  30265  bnj1446  30367  bnj1447  30368  bnj1448  30369  bnj1466  30375  bnj1467  30376  nfaltop  31257  stoweidlem21  38914  stoweidlem47  38940  nfdfat  39859
 Copyright terms: Public domain W3C validator