Let we have following axioms and modus ponens : $$(A1):(B ⇒ (C ⇒B ))$$ $$(A2):((B ⇒ (C ⇒D )) ⇒ ((B ⇒C ) ⇒ (B ⇒D )))$$ $$(A3):( ( B ⇒C) ⇒(¬C ) ⇒ (¬B ))$$

now can we prove following theorem ?

$\vdash _{H^,} $(((¬C ) ⇒ (¬B )) ⇒( B ⇒C ))

I can prove transitive law in this system.