Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-dfbi6 Structured version   Visualization version   GIF version

Theorem bj-dfbi6 31730
 Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
Assertion
Ref Expression
bj-dfbi6 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Proof of Theorem bj-dfbi6
StepHypRef Expression
1 bj-dfbi5 31729 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))
2 id 22 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → ((𝜑𝜓) → (𝜑𝜓)))
3 animorr 505 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
42, 3impbid1 214 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → ((𝜑𝜓) ↔ (𝜑𝜓)))
5 biimp 204 . . 3 (((𝜑𝜓) ↔ (𝜑𝜓)) → ((𝜑𝜓) → (𝜑𝜓)))
64, 5impbii 198 . 2 (((𝜑𝜓) → (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
71, 6bitri 263 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383 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  df-or 384  df-an 385 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator