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

Theorem impbi 197
 Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
Assertion
Ref Expression
impbi ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))

Proof of Theorem impbi
StepHypRef Expression
1 df-bi 196 . . 3 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 simprim 161 . . 3 (¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))) → (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))
43expi 160 1 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  impbii  198  impbidd  199  dfbi1  202  bj-bisym  31748  eqsbc3rVD  38097  orbi1rVD  38105  3impexpVD  38113  3impexpbicomVD  38114  imbi12VD  38131  sbcim2gVD  38133  sb5ALTVD  38171
 Copyright terms: Public domain W3C validator