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Metamath Proof Explorer

Theorem biimpr

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999) (Proof shortened by Wolf Lammen, 11-Nov-2012)

		Ref	Expression
	Assertion	biimpr	$⊢ (φ \leftrightarrow ψ) \to (ψ \to φ)$

Step	Hyp	Ref	Expression
1		dfbi1	$⊢ (φ \leftrightarrow ψ) \leftrightarrow \neg ((φ \to ψ) \to \neg (ψ \to φ))$
2		simprim	$⊢ \neg ((φ \to ψ) \to \neg (ψ \to φ)) \to (ψ \to φ)$
3	1 2	sylbi	$⊢ (φ \leftrightarrow ψ) \to (ψ \to φ)$