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

Theorem 3anorOLD 1095
Description: Obsolete version of 3anor 1098 as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
3anorOLD ((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3anorOLD
StepHypRef Expression
1 df-3an 1074 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 anor 511 . . . 4 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
3 ianor 510 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
43orbi1i 543 . . . 4 ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
52, 4xchbinx 323 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
6 df-3or 1073 . . 3 ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
75, 6xchbinxr 324 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
81, 7bitri 264 1 ((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383  w3o 1071  w3a 1072
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  df-or 384  df-an 385  df-3or 1073  df-3an 1074
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator