Theorem impimprbi 827

Description: An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 478, but ↔ is a weaker operator than ∧. Note that an implication and its reverse can never be simultaneously false, because of pm2.521 179. (Contributed by Wolf Lammen, 18-Dec-2023.)

Assertion

Ref	Expression
impimprbi	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))

Proof of Theorem impimprbi

Step	Hyp	Ref	Expression
1		dfbi2 478	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2		pm5.1 822	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
3	1, 2	sylbi 220	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
4		impbi 211	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
5		pm2.521 179	. . . 4 ⊢ (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑))
6	5	pm2.24d 154	. . 3 ⊢ (¬ (𝜑 → 𝜓) → (¬ (𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
7	4, 6	bija 385	. 2 ⊢ (((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
8	3, 7	impbii 212	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  norass  1534