Theorem imp_imp_extl ≪ | index | src | ≫

\imp左提取

theorem imp_imp_extl (A B C: wff):
  $ ((A \imp B) \imp A \imp C) \imp A \imp B \imp C $;
StepHypRefExpression
1 imp_tran 
(((A \imp B) \imp A \imp C) \imp B \imp A \imp C) \imp ((B \imp A \imp C) \imp A \imp B \imp C) \imp ((A \imp B) \imp A \imp C) \imp A \imp B \imp C
2 imp_tran 
(B \imp A \imp B) \imp ((A \imp B) \imp A \imp C) \imp B \imp A \imp C
3 introl_imp 
B \imp A \imp B
4 2, 3 ax_mp 
((A \imp B) \imp A \imp C) \imp B \imp A \imp C
5 1, 4 ax_mp 
((B \imp A \imp C) \imp A \imp B \imp C) \imp ((A \imp B) \imp A \imp C) \imp A \imp B \imp C
6 imp_imp_swapl 
(B \imp A \imp C) \imp A \imp B \imp C
7 5, 6 ax_mp 
((A \imp B) \imp A \imp C) \imp A \imp B \imp C

Axiom use

Logic (ax_mp, ax_1, ax_2)