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Theorem nfcvb 4643
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 4642 . . . 4  |-  -.  F/_ y y
2 eqidd 2462 . . . . 5  |-  ( A. x  x  =  y  ->  y  =  y )
32drnfc1 2619 . . . 4  |-  ( A. x  x  =  y  ->  ( F/_ x y  <->  F/_ y y ) )
41, 3mtbiri 309 . . 3  |-  ( A. x  x  =  y  ->  -.  F/_ x y )
54con2i 125 . 2  |-  ( F/_ x y  ->  -.  A. x  x  =  y )
6 nfcvf 2625 . 2  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
75, 6impbii 192 1  |-  ( F/_ x y  <->  -.  A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189   A.wal 1452   F/_wnfc 2589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-nul 4547  ax-pow 4594
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-cleq 2454  df-clel 2457  df-nfc 2591
This theorem is referenced by: (None)
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