Theorem bj-dfbi4

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

	Ref	Expression
Assertion	bj-dfbi4	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi3	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) )
2		pm4.56	⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∨ 𝜓 ) )
3	2	orbi2i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 ∨ 𝜓 ) ) )
4	1 3	bitri	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 ∨ 𝜓 ) ) )