MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3biant1d Structured version   Visualization version   GIF version

Theorem 3biant1d 1603
Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 528. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
Hypothesis
Ref Expression
3biantd.1 (𝜑𝜃)
Assertion
Ref Expression
3biant1d (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))

Proof of Theorem 3biant1d
StepHypRef Expression
1 3biantd.1 . . 3 (𝜑𝜃)
21biantrurd 529 . 2 (𝜑 → ((𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓))))
3 3anass 1117 . 2 ((𝜃𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓)))
42, 3syl6bbr 281 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108
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  df-an 386  df-3an 1110
This theorem is referenced by:  metuel2  22695  itgsubst  24150  clwlkclwwlk  27287  clwlkclwwlkOLD  27288  dfgcd3  33661  itg2addnclem2  33942
  Copyright terms: Public domain W3C validator