Theorem issetf 2737

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 2736	. 2 |- ( A e. _V <-> E. y y = A )
2		issetf.1	. . . 4 |- F/_ x A
3	2	nfeq2 2324	. . 3 |- F/ x y = A
4		nfv 1521	. . 3 |- F/ y x = A
5		eqeq1 2177	. . 3 |- ( y = x -> ( y = A <-> x = A ) )
6	3, 4, 5	cbvex 1749	. 2 |- ( E. y y = A <-> E. x x = A )
7	1, 6	bitri 183	1 |- ( A e. _V <-> E. x x = A )

Syntax hints:   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  vtoclgf  2788  spcimgft  2806  spcimegft  2808  bj-vtoclgft  13810