Theorem pm2.01 188

Description: Weak Clavius law. If a formula implies its negation, then it is false. A form of "reductio ad absurdum", which can be used in proofs by contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in minimal calculus, contrary to the Clavius law pm2.18 128. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.)

Assertion

Ref	Expression
pm2.01	⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Proof of Theorem pm2.01

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (¬ 𝜑 → ¬ 𝜑)
2	1, 1	ja 186	1 ⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm2.01i  189  bijust0  204  pm4.8  392  axin1  2694  dtrucor2  5352  ominf  9276  ominfOLD  9277  elirr  9619  hfninf  36162  bj-nimn  36539  bj-dtrucor2v  36793  remul01  42416