Theorem issetf 3208

Description: A version of isset 3207 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 3207	. 2 |- ( A e. _V <-> E. y y = A )
2		issetf.1	. . . 4 |- F/_ x A
3	2	nfeq2 2780	. . 3 |- F/ x y = A
4		nfv 1843	. . 3 |- F/ y x = A
5		eqeq1 2626	. . 3 |- ( y = x -> ( y = A <-> x = A ) )
6	3, 4, 5	cbvex 2272	. 2 |- ( E. y y = A <-> E. x x = A )
7	1, 6	bitri 264	1 |- ( A e. _V <-> E. x x = A )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  vtoclgf  3264  spcimgft  3284