Theorem imp_imp_distl ≪ | index | src | ≫

\imp\imp左分配

theorem imp_imp_distl (A B C: wff):
  $ A \imp B \imp C \iff (A \imp B) \imp A \imp C $;
StepHypRefExpression
1 iff_comp 
((A \imp B \imp C) \imp (A \imp B) \imp A \imp C) \imp (((A \imp B) \imp A \imp C) \imp A \imp B \imp C) \imp (A \imp B \imp C \iff (A \imp B) \imp A \imp C)
2 imp_imp_insl 
(A \imp B \imp C) \imp (A \imp B) \imp A \imp C
3 1, 2 ax_mp 
(((A \imp B) \imp A \imp C) \imp A \imp B \imp C) \imp (A \imp B \imp C \iff (A \imp B) \imp A \imp C)
4 imp_imp_extl 
((A \imp B) \imp A \imp C) \imp A \imp B \imp C
5 3, 4 ax_mp 
A \imp B \imp C \iff (A \imp B) \imp A \imp C

Axiom use

Logic (ax_mp, ax_1, ax_2, ax_3)