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

Theorem impbidd 202
 Description: Deduce an equivalence from two implications. Double deduction associated with impbi 200 and impbii 201. Deduction associated with impbid 204. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Hypotheses
Ref Expression
impbidd.1 (𝜑 → (𝜓 → (𝜒𝜃)))
impbidd.2 (𝜑 → (𝜓 → (𝜃𝜒)))
Assertion
Ref Expression
impbidd (𝜑 → (𝜓 → (𝜒𝜃)))

Proof of Theorem impbidd
StepHypRef Expression
1 impbidd.1 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
2 impbidd.2 . 2 (𝜑 → (𝜓 → (𝜃𝜒)))
3 impbi 200 . 2 ((𝜒𝜃) → ((𝜃𝜒) → (𝜒𝜃)))
41, 2, 3syl6c 70 1 (𝜑 → (𝜓 → (𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198 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 199 This theorem is referenced by:  impbid21d  203  pm5.74  262  seglecgr12  32731  prtlem18  34898
 Copyright terms: Public domain W3C validator