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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 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜓𝜑)))

Proof of Theorem impimprbi
StepHypRef Expression
1 dfbi2 478 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
2 pm5.1 822 . . 3 (((𝜑𝜓) ∧ (𝜓𝜑)) → ((𝜑𝜓) ↔ (𝜓𝜑)))
31, 2sylbi 220 . 2 ((𝜑𝜓) → ((𝜑𝜓) ↔ (𝜓𝜑)))
4 impbi 211 . . 3 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
5 pm2.521 179 . . . 4 (¬ (𝜑𝜓) → (𝜓𝜑))
65pm2.24d 154 . . 3 (¬ (𝜑𝜓) → (¬ (𝜓𝜑) → (𝜑𝜓)))
74, 6bija 385 . 2 (((𝜑𝜓) ↔ (𝜓𝜑)) → (𝜑𝜓))
83, 7impbii 212 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜓𝜑)))
 Colors of variables: wff setvar class 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
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