Theorem ibibr

Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005) (Proof shortened by Wolf Lammen, 21-Dec-2013)

	Ref	Expression
Assertion	ibibr	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜓 ↔ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.501	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )
2		bicom	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) )
3	1 2	bitrdi	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) ) )
4	3	pm5.74i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜓 ↔ 𝜑 ) ) )