Theorem pm3.24 852

Description: Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Assertion

Ref	Expression
pm3.24	⊢ ¬ (φ ∧ ¬ φ)

Proof of Theorem pm3.24

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (φ → φ)
2		iman 413	. 2 ⊢ ((φ → φ) ↔ ¬ (φ ∧ ¬ φ))
3	1, 2	mpbi 199	1 ⊢ ¬ (φ ∧ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358

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

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  pm4.43  893  nonconne  2524  pssirr  3370  indifdir  3512  dfnul2  3553  dfnul3  3554  rabnc  3575  nincompl  4073  imadif  5172