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Theorem issetf 2921
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 2920 . 2  |-  ( A  e.  _V  <->  E. y 
y  =  A )
2 issetf.1 . . . 4  |-  F/_ x A
32nfeq2 2551 . . 3  |-  F/ x  y  =  A
4 nfv 1626 . . 3  |-  F/ y  x  =  A
5 eqeq1 2410 . . 3  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
63, 4, 5cbvex 2038 . 2  |-  ( E. y  y  =  A  <->  E. x  x  =  A )
71, 6bitri 241 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    = wceq 1649    e. wcel 1721   F/_wnfc 2527   _Vcvv 2916
This theorem is referenced by:  vtoclgf  2970  spcimgft  2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918
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