Theorem biorfi

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995) (Proof shortened by Wolf Lammen, 16-Jul-2021)

	Ref	Expression
Hypothesis	biorfi.1	⊢ ¬ 𝜑
Assertion	biorfi	⊢ ( 𝜓 ↔ ( 𝜓 ∨ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		biorfi.1	⊢ ¬ 𝜑
2		orc	⊢ ( 𝜓 → ( 𝜓 ∨ 𝜑 ) )
3		pm2.53	⊢ ( ( 𝜓 ∨ 𝜑 ) → ( ¬ 𝜓 → 𝜑 ) )
4	1 3	mt3i	⊢ ( ( 𝜓 ∨ 𝜑 ) → 𝜓 )
5	2 4	impbii	⊢ ( 𝜓 ↔ ( 𝜓 ∨ 𝜑 ) )