Theorem tsna1

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

	Ref	Expression
Assertion	tsna1	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ⊼ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tsan1	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
2		notnotb	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ¬ ( 𝜑 ⊼ 𝜓 ) )
3		df-nan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
4	2 3	bitr3i	⊢ ( ¬ ¬ ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
5	4	con4bii	⊢ ( ¬ ( 𝜑 ⊼ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) )
6	5	orbi2i	⊢ ( ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ⊼ 𝜓 ) ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
7	1 6	sylibr	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ⊼ 𝜓 ) ) )