Theorem bj-bibibi

Description: A property of the biconditional. (Contributed by BJ, 26-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-bibibi	⊢ ( 𝜑 ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.501	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )
2		bianir	⊢ ( ( 𝜓 ∧ ( 𝜑 ↔ 𝜓 ) ) → 𝜑 )
3	2	ex	⊢ ( 𝜓 → ( ( 𝜑 ↔ 𝜓 ) → 𝜑 ) )
4		bibif	⊢ ( ¬ 𝜓 → ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ 𝜑 ) )
5	4	con2bid	⊢ ( ¬ 𝜓 → ( 𝜑 ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) )
6	5	biimprd	⊢ ( ¬ 𝜓 → ( ¬ ( 𝜑 ↔ 𝜓 ) → 𝜑 ) )
7	3 6	bija	⊢ ( ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) → 𝜑 )
8	1 7	impbii	⊢ ( 𝜑 ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜓 ) ) )