Theorem issetf 2606

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.

Step	Hyp	Ref	Expression
1		isset 2605	. 2 |- ( A e. _V <-> E. y y = A )
2		issetf.1	. . . 4 |- F/_ x A
3	2	nfeq2 2230	. . 3 |- F/ x y = A
4		nfv 1461	. . 3 |- F/ y x = A
5		eqeq1 2087	. . 3 |- ( y = x -> ( y = A <-> x = A ) )
6	3, 4, 5	cbvex 1679	. 2 |- ( E. y y = A <-> E. x x = A )
7	1, 6	bitri 182	1 |- ( A e. _V <-> E. x x = A )

Syntax hints:   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  vtoclgf  2657  spcimgft  2674  spcimegft  2676  bj-vtoclgft  10585