Theorem notnot1 559

Description: Adding double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds intuitionistically, but its converse does not (see notnot2dc 750). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)

Assertion

Ref	Expression
notnot1	⊢ (φ → ¬ ¬ φ)

Proof of Theorem notnot1

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (¬ φ → ¬ φ)
2	1	con2i 557	1 ⊢ (φ → ¬ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 544  ax-in2 545

This theorem is referenced by:  con3d  560  notnoti  573  pm3.24  626  notnotnot  627  biortn  663  dcn  745  con1dc  752  notnotdc  765  eueq2dc  2708  difsnpssim  3498  xrlttri3  8488  nltpnft  8500  ngtmnft  8501  bdnthALT  9290