MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfbi1 Structured version   Visualization version   GIF version

Theorem dfbi1 215
Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 216 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 209 . 2 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 impbi 210 . . 3 (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ((¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)) → ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))))
32con3rr3 158 . 2 (¬ ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))))
41, 3mt3 203 1 ((𝜑𝜓) ↔ ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208
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 209
This theorem is referenced by:  biimpr  222  dfbi2  477  tbw-bijust  1695  rb-bijust  1746  axrepprim  32923  axacprim  32928
  Copyright terms: Public domain W3C validator