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Theorem issetf 2967
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 2966 . 2  |-  ( A  e.  _V  <->  E. y 
y  =  A )
2 issetf.1 . . . 4  |-  F/_ x A
32nfeq2 2589 . . 3  |-  F/ x  y  =  A
4 nfv 1630 . . 3  |-  F/ y  x  =  A
5 eqeq1 2448 . . 3  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
63, 4, 5cbvex 1986 . 2  |-  ( E. y  y  =  A  <->  E. x  x  =  A )
71, 6bitri 242 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1727   F/_wnfc 2565   _Vcvv 2962
This theorem is referenced by:  vtoclgf  3016  spcimgft  3033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964
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