Theorem issetf 2779

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 2778	. 2 |- ( A e. _V <-> E. y y = A )
2		issetf.1	. . . 4 |- F/_ x A
3	2	nfeq2 2360	. . 3 |- F/ x y = A
4		nfv 1551	. . 3 |- F/ y x = A
5		eqeq1 2212	. . 3 |- ( y = x -> ( y = A <-> x = A ) )
6	3, 4, 5	cbvex 1779	. 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 1373   E.wex 1515   e. wcel 2176   F/_wnfc 2335   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  vtoclgf  2831  spcimgft  2849  spcimegft  2851  bj-vtoclgft  15748