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Metamath Proof Explorer

Theorem bnj345

Description: /\ -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 14-Jun-2011) (New usage is discouraged.)

		Ref	Expression
	Assertion	bnj345	$⊢ (φ \land ψ \land χ \land θ) \leftrightarrow (θ \land φ \land ψ \land χ)$

Proof

Step	Hyp	Ref	Expression
1		bnj334	$⊢ (φ \land ψ \land χ \land θ) \leftrightarrow (χ \land φ \land ψ \land θ)$
2		bnj250	$⊢ (χ \land φ \land ψ \land θ) \leftrightarrow (χ \land ((φ \land ψ) \land θ))$
3		3anass	$⊢ (χ \land (φ \land ψ) \land θ) \leftrightarrow (χ \land ((φ \land ψ) \land θ))$
4	2 3	bitr4i	$⊢ (χ \land φ \land ψ \land θ) \leftrightarrow (χ \land (φ \land ψ) \land θ)$
5		3anrev	$⊢ (χ \land (φ \land ψ) \land θ) \leftrightarrow (θ \land (φ \land ψ) \land χ)$
6		bnj250	$⊢ (θ \land φ \land ψ \land χ) \leftrightarrow (θ \land ((φ \land ψ) \land χ))$
7		3anass	$⊢ (θ \land (φ \land ψ) \land χ) \leftrightarrow (θ \land ((φ \land ψ) \land χ))$
8	6 7	bitr4i	$⊢ (θ \land φ \land ψ \land χ) \leftrightarrow (θ \land (φ \land ψ) \land χ)$
9	5 8	bitr4i	$⊢ (χ \land (φ \land ψ) \land θ) \leftrightarrow (θ \land φ \land ψ \land χ)$
10	1 4 9	3bitri	$⊢ (φ \land ψ \land χ \land θ) \leftrightarrow (θ \land φ \land ψ \land χ)$