Metamath Proof Explorer

Theorem bian1d

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017)

		Ref	Expression
	Hypothesis	bian1d.1	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
	Assertion	bian1d	$⊢ φ \to ((χ \land ψ) \leftrightarrow (χ \land θ))$

Step	Hyp	Ref	Expression
1		bian1d.1	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
2	1	biimpd	$⊢ φ \to (ψ \to (χ \land θ))$
3	2	adantld	$⊢ φ \to ((χ \land ψ) \to (χ \land θ))$
4		simpl	$⊢ (χ \land θ) \to χ$
5	4	a1i	$⊢ φ \to ((χ \land θ) \to χ)$
6	1	biimprd	$⊢ φ \to ((χ \land θ) \to ψ)$
7	5 6	jcad	$⊢ φ \to ((χ \land θ) \to (χ \land ψ))$
8	3 7	impbid	$⊢ φ \to ((χ \land ψ) \leftrightarrow (χ \land θ))$