Theorem ts3an1 36235

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

Assertion

Ref	Expression
ts3an1	⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒)))

Proof of Theorem ts3an1

Step	Hyp	Ref	Expression
1		tsan1 36226	. 2 ⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
2		df-3an 1087	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
3	2	orbi2i 909	. 2 ⊢ (((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒)) ↔ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
4	1, 3	sylibr 233	1 ⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 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-an 396  df-or 844  df-3an 1087

This theorem is referenced by: (None)