Theorem issetf 2807

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 2806	. 2 |- ( A e. _V <-> E. y y = A )
2		issetf.1	. . . 4 |- F/_ x A
3	2	nfeq2 2384	. . 3 |- F/ x y = A
4		nfv 1574	. . 3 |- F/ y x = A
5		eqeq1 2236	. . 3 |- ( y = x -> ( y = A <-> x = A ) )
6	3, 4, 5	cbvex 1802	. 2 |- ( E. y y = A <-> E. x x = A )
7	1, 6	bitri 184	1 |- ( A e. _V <-> E. x x = A )

Syntax hints:   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   F/_wnfc 2359   _Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  vtoclgf  2859  spcimgft  2879  spcimegft  2881  bj-vtoclgft  16139