Theorem bian1d

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017) (Proof shortened by Hongxiu Chen, 29-Jun-2025)

	Ref	Expression
Hypothesis	bian1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
Assertion	bian1d	$$⊢\phi \to \left(\left(\chi \wedge \psi \right)\leftrightarrow \left(\chi \wedge \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bian1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
2		ibar	$$⊢\chi \to \left(\theta \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
3	2	bicomd	$$⊢\chi \to \left(\left(\chi \wedge \theta \right)\leftrightarrow \theta \right)$$
4	1 3	sylan9bb	$$⊢\left(\phi \wedge \chi \right)\to \left(\psi \leftrightarrow \theta \right)$$
5	4	ex	$$⊢\phi \to \left(\chi \to \left(\psi \leftrightarrow \theta \right)\right)$$
6	5	pm5.32d	$$⊢\phi \to \left(\left(\chi \wedge \psi \right)\leftrightarrow \left(\chi \wedge \theta \right)\right)$$