Theorem bj-dfbi6

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

	Ref	Expression
Assertion	bj-dfbi6	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-dfbi5	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) )
2		id	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) )
3		animorr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∨ 𝜓 ) )
4	2 3	impbid1	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )
5		biimp	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) → ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) )
6	4 5	impbii	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )
7	1 6	bitri	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )