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Theorem dfbi1 216
 Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 217 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 210 . 2 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 impbi 211 . . 3 (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ((¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)) → ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))))
32con3rr3 158 . 2 (¬ ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))))
41, 3mt3 204 1 ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209 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 This theorem is referenced by:  biimpr  223  dfbi2  479  tbw-bijust  1701  rb-bijust  1752  axrepprim  33144  axacprim  33149
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