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Theorem issetf 2793
Description: A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypothesis
Ref Expression
issetf.1  |-  F/_ x A
Assertion
Ref Expression
issetf  |-  ( A  e.  _V  <->  E. x  x  =  A )

Proof of Theorem issetf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 isset 2792 . 2  |-  ( A  e.  _V  <->  E. y 
y  =  A )
2 issetf.1 . . . 4  |-  F/_ x A
32nfeq2 2430 . . 3  |-  F/ x  y  =  A
4 nfv 1605 . . 3  |-  F/ y  x  =  A
5 eqeq1 2289 . . 3  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
63, 4, 5cbvex 1925 . 2  |-  ( E. y  y  =  A  <->  E. x  x  =  A )
71, 6bitri 240 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788
This theorem is referenced by:  vtoclgf  2842  spcimgft  2859
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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