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

Theorem nfnfc1 2905
 Description: The setvar 𝑥 is bound in Ⅎ𝑥𝐴. (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 2891 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfnf1 2180 . . 3 𝑥𝑥 𝑦𝐴
32nfal 2300 . 2 𝑥𝑦𝑥 𝑦𝐴
41, 3nfxfr 1928 1 𝑥𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  ∀wal 1630  Ⅎwnf 1857   ∈ wcel 2139  Ⅎwnfc 2889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-nfc 2891 This theorem is referenced by:  vtoclgft  3394  vtoclgftOLD  3395  sbcralt  3651  sbcrext  3652  sbcrextOLD  3653  csbiebt  3694  nfopd  4570  nfimad  5633  nffvd  6361  nfded  34757  nfded2  34758  nfunidALT2  34759
 Copyright terms: Public domain W3C validator