Theorem adh-minimp-sylsimp 48009

Description: Derivation of jarr 107 (also called "syll-simp") from minimp 1654 and ax-mp 5. Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-sylsimp	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Proof of Theorem adh-minimp-sylsimp

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-ax2c-lem3 48008	. . 3 ⊢ (((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒))
2		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . . 4 ⊢ (((𝜑 → 𝜓) → (((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒))) → (((((𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜓)) → ((((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))))
3		adh-minimp-jarr-lem2 48007	. . . 4 ⊢ ((((𝜑 → 𝜓) → (((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒))) → (((((𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜓)) → ((((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)))) → ((((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))))
4	2, 3	ax-mp 5	. . 3 ⊢ ((((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((((𝜑 → 𝜓) → (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → 𝜒)) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)))
5	1, 4	ax-mp 5	. 2 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))
6		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))) → (((𝜓 → ((𝜑 → 𝜓) → 𝜒)) → (((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)) → (𝜓 → 𝜒))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))))
7		adh-minimp-jarr-lem2 48007	. . 3 ⊢ (((((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))) → (((𝜓 → ((𝜑 → 𝜓) → 𝜒)) → (((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)) → (𝜓 → 𝜒))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))) → (((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))))
8	6, 7	ax-mp 5	. 2 ⊢ (((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
9	5, 8	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  adh-minimp-ax1  48010  adh-minimp-imim1  48011  adh-minimp-ax2c  48012  adh-minimp-ax2-lem4  48013