Theorem tsim1

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

	Ref	Expression
Assertion	tsim1	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exmid	⊢ ( ( 𝜑 → 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) )
2		df-or	⊢ ( ( ¬ 𝜑 ∨ 𝜓 ) ↔ ( ¬ ¬ 𝜑 → 𝜓 ) )
3		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
4	3	bicomi	⊢ ( ¬ ¬ 𝜑 ↔ 𝜑 )
5	4	imbi1i	⊢ ( ( ¬ ¬ 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜓 ) )
6	2 5	bitri	⊢ ( ( ¬ 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 → 𝜓 ) )
7	6	orbi1i	⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) ) )
8	1 7	mpbir	⊢ ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) )
9	8	a1i	⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜓 ) ) )