Theorem tsbi1

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

	Ref	Expression
Assertion	tsbi1	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	olcd	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )
3		pm3.13	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
4	3	orcd	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )
5	2 4	pm2.61i	⊢ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) )
6	5	a1i	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )