Theorem dfnot 1262

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 1250	. 2 |- -. F.
2		mtt 610	. 2 |- ( -. F. -> ( -. ph <-> ( ph -> F. ) ) )
3	1, 2	ax-mp 7	1 |- ( -. ph <-> ( ph -> F. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   F. wfal 1248

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  inegd  1263  pclem6  1265  alnex  1388  alexim  1536  difin  3174  indifdir  3193  recvguniq  9593  bj-axempty2  10014