Theorem adh-jarrsc 44446

Description: Replacement of a nested antecedent with an outer antecedent. Commuted simplificated form of elimination of a nested antecedent. Also holds intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
adh-jarrsc	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜃 → (𝜓 → 𝜒)))

Proof of Theorem adh-jarrsc

Step	Hyp	Ref	Expression
1		jarr 106	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
2		ax-1 6	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) → (𝜃 → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))))
3	1, 2	ax-mp 5	. 2 ⊢ (𝜃 → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
4		pm2.04 90	. 2 ⊢ ((𝜃 → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))) → (((𝜑 → 𝜓) → 𝜒) → (𝜃 → (𝜓 → 𝜒))))
5	3, 4	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜃 → (𝜓 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  adh-minim  44447