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Theorem nfcvb 4205
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 4204 . . . 4  |-  -.  F/_ y y
2 eqidd 2284 . . . . 5  |-  ( A. x  x  =  y  ->  y  =  y )
32drnfc1 2435 . . . 4  |-  ( A. x  x  =  y  ->  ( F/_ x y  <->  F/_ y y ) )
41, 3mtbiri 294 . . 3  |-  ( A. x  x  =  y  ->  -.  F/_ x y )
54con2i 112 . 2  |-  ( F/_ x y  ->  -.  A. x  x  =  y )
6 nfcvf 2441 . 2  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
75, 6impbii 180 1  |-  ( F/_ x y  <->  -.  A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1527    = wceq 1623   F/_wnfc 2406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408
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