impimprbi

Description: An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 , but <-> is a weaker operator than /\ . Note that an implication and its reverse can never be simultaneously false, because of pm2.521 . (Contributed by Wolf Lammen, 18-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		dfbi2	$⊢ (φ \leftrightarrow ψ) \leftrightarrow ((φ \to ψ) \land (ψ \to φ))$
2		pm5.1	$⊢ ((φ \to ψ) \land (ψ \to φ)) \to ((φ \to ψ) \leftrightarrow (ψ \to φ))$
3	1 2	sylbi	$⊢ (φ \leftrightarrow ψ) \to ((φ \to ψ) \leftrightarrow (ψ \to φ))$
4		impbi	$⊢ (φ \to ψ) \to ((ψ \to φ) \to (φ \leftrightarrow ψ))$
5		pm2.521	$⊢ \neg (φ \to ψ) \to (ψ \to φ)$
6	5	pm2.24d	$⊢ \neg (φ \to ψ) \to (\neg (ψ \to φ) \to (φ \leftrightarrow ψ))$
7	4 6	bija	$⊢ ((φ \to ψ) \leftrightarrow (ψ \to φ)) \to (φ \leftrightarrow ψ)$
8	3 7	impbii	$⊢ (φ \leftrightarrow ψ) \leftrightarrow ((φ \to ψ) \leftrightarrow (ψ \to φ))$

Metamath Proof Explorer

Theorem impimprbi

Proof