Theorem nfcvb 4643

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 4642	. . . 4 |- -. F/_ y y
2		eqidd 2462	. . . . 5 |- ( A. x x = y -> y = y )
3	2	drnfc1 2619	. . . 4 |- ( A. x x = y -> ( F/_ x y <-> F/_ y y ) )
4	1, 3	mtbiri 309	. . 3 |- ( A. x x = y -> -. F/_ x y )
5	4	con2i 125	. 2 |- ( F/_ x y -> -. A. x x = y )
6		nfcvf 2625	. 2 |- ( -. A. x x = y -> F/_ x y )
7	5, 6	impbii 192	1 |- ( F/_ x y <-> -. A. x x = y )

Syntax hints:   -. wn 3   <-> wb 189   A.wal 1452   F/_wnfc 2589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-nul 4547  ax-pow 4594

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-cleq 2454  df-clel 2457  df-nfc 2591

This theorem is referenced by: (None)