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)

		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		anass	$⊢ ((φ \land ψ) \land (χ \land θ)) \leftrightarrow (φ \land (ψ \land (χ \land θ)))$
2		anass	$⊢ ((φ \land (χ \land θ)) \land ψ) \leftrightarrow (φ \land ((χ \land θ) \land ψ))$
3		3anass	$⊢ (φ \land χ \land θ) \leftrightarrow (φ \land (χ \land θ))$
4	3	anbi1i	$⊢ ((φ \land χ \land θ) \land ψ) \leftrightarrow ((φ \land (χ \land θ)) \land ψ)$
5		ancom	$⊢ (ψ \land (χ \land θ)) \leftrightarrow ((χ \land θ) \land ψ)$
6	5	anbi2i	$⊢ (φ \land (ψ \land (χ \land θ))) \leftrightarrow (φ \land ((χ \land θ) \land ψ))$
7	2 4 6	3bitr4ri	$⊢ (φ \land (ψ \land (χ \land θ))) \leftrightarrow ((φ \land χ \land θ) \land ψ)$
8	1 7	bitri	$⊢ ((φ \land ψ) \land (χ \land θ)) \leftrightarrow ((φ \land χ \land θ) \land ψ)$