Theorem impimprbi 825

Description: An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 474, but ↔ is a weaker operator than ∧. Note that an implication and its reverse can never be simultaneously false, because of pm2.521 176. (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 474	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2		pm5.1 820	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
3	1, 2	sylbi 216	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
4		impbi 207	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
5		pm2.521 176	. . . 4 ⊢ (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑))
6	5	pm2.24d 151	. . 3 ⊢ (¬ (𝜑 → 𝜓) → (¬ (𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
7	4, 6	bija 381	. 2 ⊢ (((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
8	3, 7	impbii 208	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  norass  1535