Theorem ts3an2 38833

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

Assertion

Ref	Expression
ts3an2	⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))

Proof of Theorem ts3an2

Step	Hyp	Ref	Expression
1		tsan2 38824	. 2 ⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
2		df-3an 1105	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
3	2	notbii 323	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ ((𝜑 ∧ 𝜓) ∧ 𝜒))
4	3	orbi2i 926	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
5	1, 4	sylibr 237	1 ⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105

This theorem is used by: (None)