Theorem ts3an2

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

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

Proof

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