Theorem imp_and_distl ≪ | index | src | ≫

\imp\and左分配

theorem imp_and_distl (A B C: wff):
  $ A \imp B \and C \iff (A \imp B) \and (A \imp C) $;
StepHypRefExpression
1 iff_comp 
((A \imp B \and C) \imp (A \imp B) \and (A \imp C)) \imp
  ((A \imp B) \and (A \imp C) \imp A \imp B \and C) \imp
  (A \imp B \and C \iff (A \imp B) \and (A \imp C))
2 imp_and_insl 
(A \imp B \and C) \imp (A \imp B) \and (A \imp C)
3 1, 2 ax_mp 
((A \imp B) \and (A \imp C) \imp A \imp B \and C) \imp (A \imp B \and C \iff (A \imp B) \and (A \imp C))
4 imp_and_extl 
(A \imp B) \and (A \imp C) \imp A \imp B \and C
5 3, 4 ax_mp 
A \imp B \and C \iff (A \imp B) \and (A \imp C)

Axiom use

Logic (ax_mp, ax_1, ax_2, ax_3)