Theorem nfcvb 4686

Description: The "distinctor" expression -. A. x x = y, stating that x and y are not the same variable, can be written in terms of F/ in the obvious way. This theorem is not true in a one-element domain, because then F/_ x y and A. x x = y will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)

Assertion

Ref	Expression
nfcvb	|- ( F/_ x y <-> -. A. x x = y )

Distinct variable group:   x, y

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 4685	. . . 4 |- -. F/_ y y
2		eqidd 2458	. . . . 5 |- ( A. x x = y -> y = y )
3	2	drnfc1 2638	. . . 4 |- ( A. x x = y -> ( F/_ x y <-> F/_ y y ) )
4	1, 3	mtbiri 303	. . 3 |- ( A. x x = y -> -. F/_ x y )
5	4	con2i 120	. 2 |- ( F/_ x y -> -. A. x x = y )
6		nfcvf 2644	. 2 |- ( -. A. x x = y -> F/_ x y )
7	5, 6	impbii 188	1 |- ( F/_ x y <-> -. A. x x = y )

Syntax hints:   -. wn 3   <-> wb 184   A.wal 1393   F/_wnfc 2605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586  ax-pow 4634

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-cleq 2449  df-clel 2452  df-nfc 2607

This theorem is referenced by: (None)