Theorem ts3or2

Description: A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018)

	Ref	Expression
Assertion	ts3or2	⊢ ( 𝜃 → ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tsor2	⊢ ( 𝜃 → ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ) )
2		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
3	2	orbi2i	⊢ ( ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ) )
4	1 3	sylibr	⊢ ( 𝜃 → ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ) )