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Theorem wl-3xorbi 34849
Description: Triple xor can be replaced with a triple biconditional. Unlike , you cannot add more inputs by simply stacking up more biconditionals, and still express an "odd number of inputs". (Contributed by Wolf Lammen, 24-Apr-2024.)
Assertion
Ref Expression
wl-3xorbi (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓𝜒)))

Proof of Theorem wl-3xorbi
StepHypRef Expression
1 wl-df3xor2 34845 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
2 df-xor 1503 . 2 ((𝜑 ⊻ (𝜓𝜒)) ↔ ¬ (𝜑 ↔ (𝜓𝜒)))
3 xor3 387 . . 3 (¬ (𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ↔ ¬ (𝜓𝜒)))
4 xnor 1504 . . . 4 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
54bibi2i 341 . . 3 ((𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ↔ ¬ (𝜓𝜒)))
63, 5bitr4i 281 . 2 (¬ (𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ↔ (𝜓𝜒)))
71, 2, 63bitri 300 1 (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wxo 1502  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-3xorbi2  34850  wl-3xorrot  34853  wl-3xornot1  34856
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