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Theorem nfcvb 4394
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 4393 . . . 4  |-  -.  F/_ y y
2 eqidd 2437 . . . . 5  |-  ( A. x  x  =  y  ->  y  =  y )
32drnfc1 2588 . . . 4  |-  ( A. x  x  =  y  ->  ( F/_ x y  <->  F/_ y y ) )
41, 3mtbiri 295 . . 3  |-  ( A. x  x  =  y  ->  -.  F/_ x y )
54con2i 114 . 2  |-  ( F/_ x y  ->  -.  A. x  x  =  y )
6 nfcvf 2594 . 2  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
75, 6impbii 181 1  |-  ( F/_ x y  <->  -.  A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   F/_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
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