MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfnfc1 Structured version   Visualization version   GIF version

Theorem nfnfc1 2982
Description: The setvar 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
nfnfc1 𝑥𝑥𝐴

Proof of Theorem nfnfc1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2962 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfnf1 2158 . . 3 𝑥𝑥 𝑦𝐴
32nfal 2343 . 2 𝑥𝑦𝑥 𝑦𝐴
41, 3nfxfr 1854 1 𝑥𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wal 1536  wnf 1785  wcel 2114  wnfc 2960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-nfc 2962
This theorem is referenced by:  vtoclgft  3528  vtoclgftOLD  3529  sbcralt  3828  sbcrext  3829  csbiebt  3884  nfopd  4795  nfimad  5916  nffvd  6664  wl-dfrmof  34982  wl-dfrabf  34991  nfded  36225  nfded2  36226  nfunidALT2  36227
  Copyright terms: Public domain W3C validator