Theorem wl-orel12

Description: In a conjunctive normal form a pair of nodes like ( ph \/ ps ) /\ ( -. ph \/ ch ) eliminates the need of a node ( ps \/ ch ) . This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020)

	Ref	Expression
Assertion	wl-orel12	$$⊢\left(\left(\phi \vee \psi \right)\wedge \left(\neg \phi \vee \chi \right)\right)\to \left(\psi \vee \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.1	$$⊢\left(\neg \phi \vee \phi \right)$$
2		orel1	$$⊢\neg \phi \to \left(\left(\phi \vee \psi \right)\to \psi \right)$$
3		orc	$$⊢\psi \to \left(\psi \vee \chi \right)$$
4	2 3	syl6com	$$⊢\left(\phi \vee \psi \right)\to \left(\neg \phi \to \left(\psi \vee \chi \right)\right)$$
5		notnot	$$⊢\phi \to \neg \neg \phi$$
6		orel1	$$⊢\neg \neg \phi \to \left(\left(\neg \phi \vee \chi \right)\to \chi \right)$$
7	5 6	syl	$$⊢\phi \to \left(\left(\neg \phi \vee \chi \right)\to \chi \right)$$
8		olc	$$⊢\chi \to \left(\psi \vee \chi \right)$$
9	7 8	syl6com	$$⊢\left(\neg \phi \vee \chi \right)\to \left(\phi \to \left(\psi \vee \chi \right)\right)$$
10	4 9	jaao	$$⊢\left(\left(\phi \vee \psi \right)\wedge \left(\neg \phi \vee \chi \right)\right)\to \left(\left(\neg \phi \vee \phi \right)\to \left(\psi \vee \chi \right)\right)$$
11	1 10	mpi	$$⊢\left(\left(\phi \vee \psi \right)\wedge \left(\neg \phi \vee \chi \right)\right)\to \left(\psi \vee \chi \right)$$