Theorem adh-minimp-ax2c 48012

Description: Derivation of a commuted form of ax-2 7 from adh-minimp 48005 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (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-ax2c	⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))

Proof of Theorem adh-minimp-ax2c

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-ax2c-lem3 48008	. . . . . 6 ⊢ ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → 𝜑)
2		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . . . . 6 ⊢ (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → 𝜑) → ((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑)) → (𝜑 → (𝜓 → 𝜒))) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒))))
3	1, 2	ax-mp 5	. . . . 5 ⊢ ((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑)) → (𝜑 → (𝜓 → 𝜒))) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)))
4		adh-minimp-sylsimp 48009	. . . . 5 ⊢ (((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑)) → (𝜑 → (𝜓 → 𝜒))) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒))))
5	3, 4	ax-mp 5	. . . 4 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)))
6		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . . 4 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒))) → ((((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) → (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))))
7	5, 6	ax-mp 5	. . 3 ⊢ ((((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) → (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
8		adh-minimp-sylsimp 48009	. . 3 ⊢ (((((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) → (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))))
9	7, 8	ax-mp 5	. 2 ⊢ ((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
10		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . 3 ⊢ ((𝜑 → 𝜓) → (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
11		adh-minimp-imim1 48011	. . 3 ⊢ (((𝜑 → 𝜓) → (((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))))
12	10, 11	ax-mp 5	. 2 ⊢ (((((((𝜃 → 𝜏) → (((𝜂 → 𝜃) → (𝜏 → 𝜁)) → (𝜃 → 𝜁))) → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))))
13	9, 12	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 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-ax2-lem4  48013  adh-minimp-ax2  48014