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Theorem nfcvb 4686
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 4685 . . . 4  |-  -.  F/_ y y
2 eqidd 2458 . . . . 5  |-  ( A. x  x  =  y  ->  y  =  y )
32drnfc1 2638 . . . 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 2644 . 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 1393   F/_wnfc 2605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586  ax-pow 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-cleq 2449  df-clel 2452  df-nfc 2607
This theorem is referenced by: (None)
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