Theorem dfnot 1302

Description: Given falsum, we can define the negation of a wff ph as the statement that a contradiction follows from assuming ph. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)

Assertion

Ref	Expression
dfnot	|- ( -. ph <-> ( ph -> F. ) )

Proof of Theorem dfnot

Step	Hyp	Ref	Expression
1		fal 1291	. 2 |- -. F.
2		mtt 642	. 2 |- ( -. F. -> ( -. ph <-> ( ph -> F. ) ) )
3	1, 2	ax-mp 7	1 |- ( -. ph <-> ( ph -> F. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   F. wfal 1289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  inegd  1303  pclem6  1305  alnex  1428  alexim  1576  difin  3201  indifdir  3220  recvguniq  9881  bj-axempty2  10685