Theorem wl-orel12 34753

Description: In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)

Assertion

Ref	Expression
wl-orel12	⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒))

Proof of Theorem wl-orel12

Step	Hyp	Ref	Expression
1		pm2.1 893	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
2		orel1 885	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3		orc 863	. . . 4 ⊢ (𝜓 → (𝜓 ∨ 𝜒))
4	2, 3	syl6com 37	. . 3 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → (𝜓 ∨ 𝜒)))
5		notnot 144	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
6		orel1 885	. . . . 5 ⊢ (¬ ¬ 𝜑 → ((¬ 𝜑 ∨ 𝜒) → 𝜒))
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → ((¬ 𝜑 ∨ 𝜒) → 𝜒))
8		olc 864	. . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒))
9	7, 8	syl6com 37	. . 3 ⊢ ((¬ 𝜑 ∨ 𝜒) → (𝜑 → (𝜓 ∨ 𝜒)))
10	4, 9	jaao 951	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → ((¬ 𝜑 ∨ 𝜑) → (𝜓 ∨ 𝜒)))
11	1, 10	mpi 20	1 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844

This theorem is referenced by:  wl-cases2-dnf  34754