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Metamath Proof Explorer

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	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
	Assertion	bian1d	$⊢ φ \to ((χ \land ψ) \leftrightarrow (χ \land θ))$

Proof

Step	Hyp	Ref	Expression
1		bian1d.1	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
2		ibar	$⊢ χ \to (θ \leftrightarrow (χ \land θ))$
3	2	bicomd	$⊢ χ \to ((χ \land θ) \leftrightarrow θ)$
4	1 3	sylan9bb	$⊢ (φ \land χ) \to (ψ \leftrightarrow θ)$
5	4	ex	$⊢ φ \to (χ \to (ψ \leftrightarrow θ))$
6	5	pm5.32d	$⊢ φ \to ((χ \land ψ) \leftrightarrow (χ \land θ))$