Theorem nfcvb 4898

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 )

Proof of Theorem nfcvb

Step	Hyp	Ref	Expression
1		nfnid 4897	. . . 4 |- -. F/_ y y
2		eqidd 2623	. . . . 5 |- ( A. x x = y -> y = y )
3	2	drnfc1 2782	. . . 4 |- ( A. x x = y -> ( F/_ x y <-> F/_ y y ) )
4	1, 3	mtbiri 317	. . 3 |- ( A. x x = y -> -. F/_ x y )
5	4	con2i 134	. 2 |- ( F/_ x y -> -. A. x x = y )
6		nfcvf 2788	. 2 |- ( -. A. x x = y -> F/_ x y )
7	5, 6	impbii 199	1 |- ( F/_ x y <-> -. A. x x = y )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   F/_wnfc 2751

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

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-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)