Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imbi13 Structured version   Visualization version   GIF version

Theorem imbi13 38546
Description: Join three logical equivalences to form equivalence of implications. imbi13 38546 is imbi13VD 38930 without virtual deductions and was automatically derived from imbi13VD 38930 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi13 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Proof of Theorem imbi13
StepHypRef Expression
1 imbi12 336 . 2 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
2 imbi12 336 . 2 ((𝜑𝜓) → (((𝜒𝜏) ↔ (𝜃𝜂)) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))
31, 2syl9r 78 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196
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 197
This theorem is referenced by:  trsbc  38570  trsbcVD  38933
  Copyright terms: Public domain W3C validator