ordi

Description: Distributive law for disjunction. Theorem *4.41 of WhiteheadRussell p. 119. (Contributed by NM, 5-Jan-1993) (Proof shortened by Andrew Salmon, 7-May-2011) (Proof shortened by Wolf Lammen, 28-Nov-2013)

		Ref	Expression
	Assertion	ordi	$⊢ (φ \lor (ψ \land χ)) \leftrightarrow ((φ \lor ψ) \land (φ \lor χ))$

Proof

Step	Hyp	Ref	Expression
1		jcab	$⊢ (\neg φ \to (ψ \land χ)) \leftrightarrow ((\neg φ \to ψ) \land (\neg φ \to χ))$
2		df-or	$⊢ (φ \lor (ψ \land χ)) \leftrightarrow (\neg φ \to (ψ \land χ))$
3		df-or	$⊢ (φ \lor ψ) \leftrightarrow (\neg φ \to ψ)$
4		df-or	$⊢ (φ \lor χ) \leftrightarrow (\neg φ \to χ)$
5	3 4	anbi12i	$⊢ ((φ \lor ψ) \land (φ \lor χ)) \leftrightarrow ((\neg φ \to ψ) \land (\neg φ \to χ))$
6	1 2 5	3bitr4i	$⊢ (φ \lor (ψ \land χ)) \leftrightarrow ((φ \lor ψ) \land (φ \lor χ))$

Metamath Proof Explorer

Theorem ordi

Proof