Theorem pm4.78

Description: Implication distributes over disjunction. Theorem *4.78 of WhiteheadRussell p. 121. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 19-Nov-2012)

	Ref	Expression
Assertion	pm4.78	$$⊢\left(\left(\phi \to \psi \right)\vee \left(\phi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orordi	$$⊢\left(\neg \phi \vee \left(\psi \vee \chi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \psi \right)\vee \left(\neg \phi \vee \chi \right)\right)$$
2		imor	$$⊢\left(\phi \to \left(\psi \vee \chi \right)\right)\leftrightarrow \left(\neg \phi \vee \left(\psi \vee \chi \right)\right)$$
3		imor	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\neg \phi \vee \psi \right)$$
4		imor	$$⊢\left(\phi \to \chi \right)\leftrightarrow \left(\neg \phi \vee \chi \right)$$
5	3 4	orbi12i	$$⊢\left(\left(\phi \to \psi \right)\vee \left(\phi \to \chi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \psi \right)\vee \left(\neg \phi \vee \chi \right)\right)$$
6	1 2 5	3bitr4ri	$$⊢\left(\left(\phi \to \psi \right)\vee \left(\phi \to \chi \right)\right)\leftrightarrow \left(\phi \to \left(\psi \vee \chi \right)\right)$$