Theorem adh-minim-pm2.43

Description: Derivation of pm2.43 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 , adh-minim-ax2 , and ax-mp . It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minim-pm2.43	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜑 ) )
2		adh-minim-ax2	⊢ ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) )
4		adh-minim-ax2	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜓 ) ) )
5		adh-minim-ax2	⊢ ( ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜑 ) ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) )
7	3 6	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) )