Theorem adh-minim-ax2c 43320

Description: Derivation of a commuted form of ax-2 7 from adh-minim 43312 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minim-ax2c	⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))

Proof of Theorem adh-minim-ax2c

Step	Hyp	Ref	Expression
1		adh-minim-ax2-lem5 43318	. 2 ⊢ ((𝜑 → 𝜓) → ((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))))
2		adh-minim-ax2-lem6 43319	. . . . 5 ⊢ (((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑)) → ((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → 𝜑))
3		adh-minim-ax2-lem6 43319	. . . . 5 ⊢ ((((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑)) → ((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → 𝜑)) → (((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑)) → 𝜑))
4	2, 3	ax-mp 5	. . . 4 ⊢ (((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑)) → 𝜑)
5		adh-minim-ax1-ax2-lem4 43316	. . . 4 ⊢ ((((((𝜎 → 𝜌) → 𝜇) → ((𝜌 → (𝜇 → 𝜆)) → (𝜌 → 𝜆))) → ((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑)) → 𝜑) → ((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → (𝜑 → 𝜓)) → (((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓)))
6	4, 5	ax-mp 5	. . 3 ⊢ ((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → (𝜑 → 𝜓)) → (((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓))
7		adh-minim-ax1-ax2-lem4 43316	. . 3 ⊢ (((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → (𝜑 → 𝜓)) → (((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓)) → (((𝜑 → 𝜓) → ((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))))
8	6, 7	ax-mp 5	. 2 ⊢ (((𝜑 → 𝜓) → ((((((𝜃 → 𝜏) → 𝜂) → ((𝜏 → (𝜂 → 𝜁)) → (𝜏 → 𝜁))) → 𝜑) → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))) → ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))))
9	1, 8	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 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-ax2  43321