Theorem dfbi

Description: Definition df-bi rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008)

	Ref	Expression
Assertion	dfbi	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ) ∧ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi2	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) )
2		dfbi2	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ) ↔ ( ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ) ∧ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) ) )
3	1 2	mpbi	⊢ ( ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ) ∧ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) → ( 𝜑 ↔ 𝜓 ) ) )