Theorem adh-minim-ax2c

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

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

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax2-lem5	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( ( ( ( 𝜃 → 𝜏 ) → 𝜂 ) → ( ( 𝜏 → ( 𝜂 → 𝜁 ) ) → ( 𝜏 → 𝜁 ) ) ) → 𝜑 ) → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) )
2		adh-minim-ax2-lem6	⊢ ( ( ( ( ( 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	⊢ ( ( ( ( ( ( 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	⊢ ( ( ( ( ( 𝜎 → 𝜌 ) → 𝜇 ) → ( ( 𝜌 → ( 𝜇 → 𝜆 ) ) → ( 𝜌 → 𝜆 ) ) ) → ( ( ( ( 𝜃 → 𝜏 ) → 𝜂 ) → ( ( 𝜏 → ( 𝜂 → 𝜁 ) ) → ( 𝜏 → 𝜁 ) ) ) → 𝜑 ) ) → 𝜑 )
5		adh-minim-ax1-ax2-lem4	⊢ ( ( ( ( ( ( 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	⊢ ( ( ( ( ( ( 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	⊢ ( ( ( ( ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) )