Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm3.24 GIF version

Theorem pm3.24 637
 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
StepHypRef Expression
1 notnot 569 . 2 (𝜑 → ¬ ¬ 𝜑)
2 imnan 634 . 2 ((𝜑 → ¬ ¬ 𝜑) ↔ ¬ (𝜑 ∧ ¬ 𝜑))
31, 2mpbi 137 1 ¬ (𝜑 ∧ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  pm4.43  867  excxor  1285  nonconne  2232  pssirr  3072  sspssn  3076  dfnul2  3254  dfnul3  3255  rabnc  3278  axnul  3910  zeoxor  10180  nnexmid  10286
 Copyright terms: Public domain W3C validator