Theorem tsbi2

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

	Ref	Expression
Assertion	tsbi2	⊢ ( 𝜃 → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.21	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	olcd	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )
3		pm4.57	⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ( 𝜑 ∨ 𝜓 ) )
4	3	biimpi	⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( 𝜑 ∨ 𝜓 ) )
5	4	orcd	⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )
6	2 5	pm2.61i	⊢ ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) )
7	6	a1i	⊢ ( 𝜃 → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) )