Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcvb Structured version   Unicode version

Theorem nfcvb 4394
 Description: The "distinctor" expression , stating that and are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then and will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
nfcvb
Distinct variable group:   ,

Proof of Theorem nfcvb
StepHypRef Expression
1 nfnid 4393 . . . 4
2 eqidd 2437 . . . . 5
32drnfc1 2588 . . . 4
41, 3mtbiri 295 . . 3
54con2i 114 . 2
6 nfcvf 2594 . 2
75, 6impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177  wal 1549  wnfc 2559 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338  ax-pow 4377 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2429  df-clel 2432  df-nfc 2561
 Copyright terms: Public domain W3C validator