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Theorem nanbi 1491
 Description: Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
Assertion
Ref Expression
nanbi ((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))

Proof of Theorem nanbi
StepHypRef Expression
1 dfbi3 1045 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 df-or 845 . . 3 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)))
3 df-nan 1483 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
43bicomi 227 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
5 nannot 1490 . . . . 5 𝜑 ↔ (𝜑𝜑))
6 nannot 1490 . . . . 5 𝜓 ↔ (𝜓𝜓))
75, 6anbi12i 629 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑𝜑) ∧ (𝜓𝜓)))
84, 7imbi12i 354 . . 3 ((¬ (𝜑𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
91, 2, 83bitri 300 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
10 nannan 1488 . 2 (((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
119, 10bitr4i 281 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ⊼ wnan 1482 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  df-or 845  df-nan 1483 This theorem is referenced by:  nic-dfim  1671  nic-dfneg  1672
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