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

Theorem nfcvb 4517
Description: The "distinctor" expression 
-.  A. x x  =  y, stating that  x and  y are not the same variable, can be written in terms of  F/ in the obvious way. This theorem is not true in a one-element domain, because then  F/_ x y and  A. x x  =  y will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
nfcvb  |-  ( F/_ x y  <->  -.  A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem nfcvb
StepHypRef Expression
1 nfnid 4516 . . . 4  |-  -.  F/_ y y
2 eqidd 2439 . . . . 5  |-  ( A. x  x  =  y  ->  y  =  y )
32drnfc1 2590 . . . 4  |-  ( A. x  x  =  y  ->  ( F/_ x y  <->  F/_ y y ) )
41, 3mtbiri 303 . . 3  |-  ( A. x  x  =  y  ->  -.  F/_ x y )
54con2i 120 . 2  |-  ( F/_ x y  ->  -.  A. x  x  =  y )
6 nfcvf 2596 . 2  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
75, 6impbii 188 1  |-  ( F/_ x y  <->  -.  A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1367   F/_wnfc 2561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416  ax-pow 4465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-cleq 2431  df-clel 2434  df-nfc 2563
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator