Theorem imbibi

Description: The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025) (Proof shortened by Wolf Lammen, 5-Jan-2025)

	Ref	Expression
Assertion	imbibi	$$⊢\left(\left(\phi \to \psi \right)\leftrightarrow \chi \right)\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.4	$$⊢\left(\phi \to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\phi \to \psi \right)$$
2		imbi2	$$⊢\left(\left(\phi \to \psi \right)\leftrightarrow \chi \right)\to \left(\left(\phi \to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
3	1 2	bitr3id	$$⊢\left(\left(\phi \to \psi \right)\leftrightarrow \chi \right)\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
4	3	pm5.74rd	$$⊢\left(\left(\phi \to \psi \right)\leftrightarrow \chi \right)\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$