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Theorem issetf 2795
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 )
Dummy variable  y is distinct from all other variables.

Proof of Theorem issetf
StepHypRef Expression
1 isset 2794 . 2  |-  ( A  e.  _V  <->  E. y 
y  =  A )
2 issetf.1 . . . 4  |-  F/_ x A
32nfeq2 2432 . . 3  |-  F/ x  y  =  A
4 nfv 1606 . . 3  |-  F/ y  x  =  A
5 eqeq1 2291 . . 3  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
63, 4, 5cbvex 1928 . 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 1529    = wceq 1624    e. wcel 1685   F/_wnfc 2408   _Vcvv 2790
This theorem is referenced by:  vtoclgf  2844  spcimgft  2861
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792
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