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	$⊢ (φ \to (φ \to ψ)) \to (φ \to ψ)$

Proof

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

Metamath Proof Explorer

Theorem adh-minim-pm2.43

Proof