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Theorem issetf 2948
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 2947 . 2  |-  ( A  e.  _V  <->  E. y 
y  =  A )
2 issetf.1 . . . 4  |-  F/_ x A
32nfeq2 2577 . . 3  |-  F/ x  y  =  A
4 nfv 1629 . . 3  |-  F/ y  x  =  A
5 eqeq1 2436 . . 3  |-  ( y  =  x  ->  (
y  =  A  <->  x  =  A ) )
63, 4, 5cbvex 1983 . 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 1550    = wceq 1652    e. wcel 1725   F/_wnfc 2553   _Vcvv 2943
This theorem is referenced by:  vtoclgf  2997  spcimgft  3014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945
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