Theorem tsbi3

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

	Ref	Expression
Assertion	tsbi3	⊢ ( 𝜃 → ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
2		con34b	⊢ ( ( 𝜓 → 𝜑 ) ↔ ( ¬ 𝜑 → ¬ 𝜓 ) )
3		pm2.54	⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜑 ∨ ¬ 𝜓 ) )
4	2 3	sylbi	⊢ ( ( 𝜓 → 𝜑 ) → ( 𝜑 ∨ ¬ 𝜓 ) )
5	1 4	syl	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ∨ ¬ 𝜓 ) )
6	5	con3i	⊢ ( ¬ ( 𝜑 ∨ ¬ 𝜓 ) → ¬ ( 𝜑 ↔ 𝜓 ) )
7	6	orri	⊢ ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) )
8	7	a1i	⊢ ( 𝜃 → ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) )