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Theorem dfbi1 203
 Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 204 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
Assertion
Ref Expression
dfbi1 ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))

Proof of Theorem dfbi1
StepHypRef Expression
1 df-bi 197 . . 3 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 simplim 163 . . 3 (¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))) → ((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))))
31, 2ax-mp 5 . 2 ((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
4 impbi 198 . . 3 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
54impi 160 . 2 (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))
63, 5impbii 199 1 ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196 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 197 This theorem is referenced by:  biimpr  210  dfbi2  661  tbw-bijust  1663  rb-bijust  1714  axrepprim  31705  axacprim  31710
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