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Theorem dfbi1 198
Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 199 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 192 . . 3 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 simplim 158 . . 3 (¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))) → ((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))))
31, 2ax-mp 5 . 2 ((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
4 impbi 193 . . 3 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
54impi 155 . 2 (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))
63, 5impbii 194 1 ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191
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 192
This theorem is referenced by:  biimpr  205  dfbi2  649  tbw-bijust  1611  rb-bijust  1662  axrepprim  30481  axacprim  30486
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