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Metamath Proof Explorer

Theorem biorf

Description: A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of WhiteheadRussell p. 121. (Contributed by NM, 23-Mar-1995) (Proof shortened by Wolf Lammen, 18-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		olc	⊢ ( 𝜓 → ( 𝜑 ∨ 𝜓 ) )
2		orel1	⊢ ( ¬ 𝜑 → ( ( 𝜑 ∨ 𝜓 ) → 𝜓 ) )
3	1 2	impbid2	⊢ ( ¬ 𝜑 → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )