Theorem dfnot 1413

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 1402	. 2 |- -. F.
2		mtt 689	. 2 |- ( -. F. -> ( -. ph <-> ( ph -> F. ) ) )
3	1, 2	ax-mp 5	1 |- ( -. ph <-> ( ph -> F. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   F. wfal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401

This theorem is referenced by:  inegd  1414  pclem6  1416  dcfrompeirce  1492  alnex  1545  alexim  1691  difin  3441  indifdir  3460  recvguniq  11501  logbgcd1irr  15635  bj-axempty2  16215