adh-minimp-imim1

Description: Derivation of imim1 ("left antimonotonicity of implication", theorem *2.06 of WhiteheadRussell p. 100) from adh-minimp and ax-mp . Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	adh-minimp-imim1	$⊢ (φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-sylsimp	$⊢ (((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ))$
2		adh-minimp-jarr-imim1-ax2c-lem1	$⊢ (φ \to ψ) \to (((θ \to φ) \to (ψ \to χ)) \to (φ \to χ))$
3		adh-minimp-jarr-imim1-ax2c-lem1	$⊢ ((φ \to ψ) \to (((θ \to φ) \to (ψ \to χ)) \to (φ \to χ))) \to (((ρ \to (φ \to ψ)) \to ((((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ)))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))))$
4	2 3	ax-mp	$⊢ ((ρ \to (φ \to ψ)) \to ((((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ)))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ)))$
5		adh-minimp-sylsimp	$⊢ (((ρ \to (φ \to ψ)) \to ((((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ)))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ)))) \to (((((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))))$
6	4 5	ax-mp	$⊢ ((((θ \to φ) \to (ψ \to χ)) \to (φ \to χ)) \to ((ψ \to χ) \to (φ \to χ))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ)))$
7	1 6	ax-mp	$⊢ (φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))$

Metamath Proof Explorer

Theorem adh-minimp-imim1

Proof