Theorem tsbi4

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

	Ref	Expression
Assertion	tsbi4	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tsbi3	⊢ ( 𝜃 → ( ( 𝜓 ∨ ¬ 𝜑 ) ∨ ¬ ( 𝜓 ↔ 𝜑 ) ) )
2		orcom	⊢ ( ( 𝜓 ∨ ¬ 𝜑 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) )
3		bicom	⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ ( 𝜑 ↔ 𝜓 ) )
4	3	notbii	⊢ ( ¬ ( 𝜓 ↔ 𝜑 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) )
5	2 4	orbi12i	⊢ ( ( ( 𝜓 ∨ ¬ 𝜑 ) ∨ ¬ ( 𝜓 ↔ 𝜑 ) ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )
6	1 5	sylib	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )