Theorem bj-dfbi5

Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019)

	Ref	Expression
Assertion	bj-dfbi5	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orcom	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 ∨ 𝜓 ) ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
2		bj-dfbi4	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 ∨ 𝜓 ) ) )
3		imor	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ¬ ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
4	1 2 3	3bitr4i	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) )