imim21b

Description: Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011) (Proof shortened by Wolf Lammen, 14-Sep-2013)

		Ref	Expression
	Assertion	imim21b	$⊢ (ψ \to φ) \to (((φ \to χ) \to (ψ \to θ)) \leftrightarrow (ψ \to (χ \to θ)))$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$⊢ ((φ \to χ) \to (ψ \to θ)) \leftrightarrow (ψ \to ((φ \to χ) \to θ))$
2		pm5.5	$⊢ φ \to ((φ \to χ) \leftrightarrow χ)$
3	2	imbi1d	$⊢ φ \to (((φ \to χ) \to θ) \leftrightarrow (χ \to θ))$
4	3	imim2i	$⊢ (ψ \to φ) \to (ψ \to (((φ \to χ) \to θ) \leftrightarrow (χ \to θ)))$
5	4	pm5.74d	$⊢ (ψ \to φ) \to ((ψ \to ((φ \to χ) \to θ)) \leftrightarrow (ψ \to (χ \to θ)))$
6	1 5	bitrid	$⊢ (ψ \to φ) \to (((φ \to χ) \to (ψ \to θ)) \leftrightarrow (ψ \to (χ \to θ)))$

Metamath Proof Explorer

Theorem imim21b

Proof