Theorem pm3.24 698

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 632	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		imnan 694	. 2 ⊢ ((𝜑 → ¬ ¬ 𝜑) ↔ ¬ (𝜑 ∧ ¬ 𝜑))
3	1, 2	mpbi 145	1 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nnexmid  855  pm4.43  955  excxor  1420  nonconne  2412  dfnul2  3493  dfnul3  3494  rabnc  3524  axnul  4209  fiintim  7101  zeoxor  12388  unennn  12976  lgsquadlem2  15765