Theorem pm3.24 627

Description: Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
pm3.24	⊢ ¬ (𝜑 ∧ ¬ 𝜑)

Proof of Theorem pm3.24

Step	Hyp	Ref	Expression
1		notnot 559	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		imnan 624	. 2 ⊢ ((𝜑 → ¬ ¬ 𝜑) ↔ ¬ (𝜑 ∧ ¬ 𝜑))
3	1, 2	mpbi 133	1 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97

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

This theorem is referenced by:  pm4.43  856  excxor  1269  nonconne  2217  pssirr  3044  sspssn  3048  dfnul2  3226  dfnul3  3227  rabnc  3250  axnul  3882  nnexmid  9899