Metamath Proof Explorer

Theorem 4anpull2

Description: An equivalence of two four-terms conjunctions with the terms regrouped (here, the second sub-conjunct of the first term is pulled separately). (Contributed by Zhi Wang, 4-Sep-2024) (Proof shortened by Garrett Katz, 26-Jun-2026)

		Ref	Expression
	Assertion	4anpull2	$⊢ ((φ \land ψ) \land (χ \land θ)) \leftrightarrow ((φ \land χ \land θ) \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		an42	$⊢ ((φ \land ψ) \land (χ \land θ)) \leftrightarrow ((φ \land χ) \land (θ \land ψ))$
2		3an4anass	$⊢ ((φ \land χ \land θ) \land ψ) \leftrightarrow ((φ \land χ) \land (θ \land ψ))$
3	1 2	bitr4i	$⊢ ((φ \land ψ) \land (χ \land θ)) \leftrightarrow ((φ \land χ \land θ) \land ψ)$