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Theorem dfbi1 184
 Description: Relate the biconditional connective to primitive connectives. See dfbi1gb 185 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
dfbi1 ((φψ) ↔ ¬ ((φψ) → ¬ (ψφ)))

Proof of Theorem dfbi1
StepHypRef Expression
1 df-bi 177 . . 3 ¬ (((φψ) → ¬ ((φψ) → ¬ (ψφ))) → ¬ (¬ ((φψ) → ¬ (ψφ)) → (φψ)))
2 simplim 143 . . 3 (¬ (((φψ) → ¬ ((φψ) → ¬ (ψφ))) → ¬ (¬ ((φψ) → ¬ (ψφ)) → (φψ))) → ((φψ) → ¬ ((φψ) → ¬ (ψφ))))
31, 2ax-mp 8 . 2 ((φψ) → ¬ ((φψ) → ¬ (ψφ)))
4 bi3 179 . . 3 ((φψ) → ((ψφ) → (φψ)))
54impi 140 . 2 (¬ ((φψ) → ¬ (ψφ)) → (φψ))
63, 5impbii 180 1 ((φψ) ↔ ¬ ((φψ) → ¬ (ψφ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177 This theorem is referenced by:  bi2  189  dfbi2  609  tbw-bijust  1463  rb-bijust  1514  nfbidOLD  1833
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