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Metamath Proof Explorer

Theorem imimorb

Description: Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012) (Proof shortened by Wolf Lammen, 3-Apr-2013)

		Ref	Expression
	Assertion	imimorb	$⊢ ((ψ \to χ) \to (φ \to χ)) \leftrightarrow (φ \to (ψ \lor χ))$

Proof

Step	Hyp	Ref	Expression
1		bi2.04	$⊢ ((ψ \to χ) \to (φ \to χ)) \leftrightarrow (φ \to ((ψ \to χ) \to χ))$
2		dfor2	$⊢ (ψ \lor χ) \leftrightarrow ((ψ \to χ) \to χ)$
3	2	imbi2i	$⊢ (φ \to (ψ \lor χ)) \leftrightarrow (φ \to ((ψ \to χ) \to χ))$
4	1 3	bitr4i	$⊢ ((ψ \to χ) \to (φ \to χ)) \leftrightarrow (φ \to (ψ \lor χ))$