Theorem ts3or2 36312

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

Assertion

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

Proof of Theorem ts3or2

Step	Hyp	Ref	Expression
1		tsor2 36306	. 2 ⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∨ 𝜒)))
2		df-3or 1087	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
3	2	orbi2i 910	. 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ ((𝜑 ∨ 𝜓) ∨ 𝜒)))
4	1, 3	sylibr 233	1 ⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   ∨ w3o 1085

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 206  df-or 845  df-3or 1087

This theorem is referenced by: (None)