Theorem biorfri

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995) (Proof shortened by Wolf Lammen, 16-Jul-2021) (Proof shortened by AV, 10-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		biorfi.1	⊢ ¬ 𝜑
2	1	biorfi	⊢ ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) )
3		orcom	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜑 ) )
4	2 3	bitri	⊢ ( 𝜓 ↔ ( 𝜓 ∨ 𝜑 ) )