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

Theorem noran 1530
Description: is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
Assertion
Ref Expression
noran ((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))

Proof of Theorem noran
StepHypRef Expression
1 anor 979 . . 3 ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))
2 nornot 1528 . . . 4 𝜑 ↔ (𝜑 𝜑))
3 nornot 1528 . . . 4 𝜓 ↔ (𝜓 𝜓))
42, 3orbi12i 911 . . 3 ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ((𝜑 𝜑) ∨ (𝜓 𝜓)))
51, 4xchbinx 333 . 2 ((𝜑𝜓) ↔ ¬ ((𝜑 𝜑) ∨ (𝜓 𝜓)))
6 df-nor 1525 . 2 (((𝜑 𝜑) (𝜓 𝜓)) ↔ ¬ ((𝜑 𝜑) ∨ (𝜓 𝜓)))
75, 6bitr4i 277 1 ((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 843   wnor 1524
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 206  df-an 396  df-or 844  df-nor 1525
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator