Theorem tsna3

Description: A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)

	Ref	Expression
Assertion	tsna3	⊢ ( 𝜃 → ( 𝜓 ∨ ( 𝜑 ⊼ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tsan3	⊢ ( 𝜃 → ( 𝜓 ∨ ¬ ( 𝜑 ∧ 𝜓 ) ) )
2		df-nan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
3	2	orbi2i	⊢ ( ( 𝜓 ∨ ( 𝜑 ⊼ 𝜓 ) ) ↔ ( 𝜓 ∨ ¬ ( 𝜑 ∧ 𝜓 ) ) )
4	1 3	sylibr	⊢ ( 𝜃 → ( 𝜓 ∨ ( 𝜑 ⊼ 𝜓 ) ) )