Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-3xorbi123d Structured version   Visualization version   GIF version

Theorem wl-3xorbi123d 34834
Description: Equivalence theorem for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
Hypotheses
Ref Expression
wl-3xorbid.1 (𝜑 → (𝜓𝜒))
wl-3xorbid.2 (𝜑 → (𝜃𝜏))
wl-3xorbid.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
wl-3xorbi123d (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))

Proof of Theorem wl-3xorbi123d
StepHypRef Expression
1 wl-3xorbid.1 . . . 4 (𝜑 → (𝜓𝜒))
2 wl-3xorbid.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2bibi12d 349 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 wl-3xorbid.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4bibi12d 349 . 2 (𝜑 → (((𝜓𝜃) ↔ 𝜂) ↔ ((𝜒𝜏) ↔ 𝜁)))
6 wl-3xorbi2 34833 . 2 (hadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ↔ 𝜂))
7 wl-3xorbi2 34833 . 2 (hadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒𝜏) ↔ 𝜁))
85, 6, 73bitr4g 317 1 (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  haddwhad 1594
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 210  df-an 400  df-or 845  df-ifp 1059  df-xor 1503  df-tru 1541  df-had 1595
This theorem is referenced by:  wl-3xorbi123i  34835
  Copyright terms: Public domain W3C validator