Theorem pm3.24 688

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 624	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		imnan 685	. 2 ⊢ ((𝜑 → ¬ ¬ 𝜑) ↔ ¬ (𝜑 ∧ ¬ 𝜑))
3	1, 2	mpbi 144	1 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nnexmid  845  pm4.43  944  excxor  1373  nonconne  2352  dfnul2  3416  dfnul3  3417  rabnc  3447  axnul  4114  fiintim  6906  zeoxor  11828  unennn  12352