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Theorem wl-df3xor2 34879
Description: Alternative definition of wl-df-3xor 34878, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a . Copy of hadass 1598. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.)
Assertion
Ref Expression
wl-df3xor2 (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))

Proof of Theorem wl-df3xor2
StepHypRef Expression
1 ifpn 1069 . 2 (if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒)) ↔ if-(¬ 𝜑, (𝜓𝜒), ¬ (𝜓𝜒)))
2 wl-df-3xor 34878 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒)))
3 df-xor 1503 . . 3 ((𝜑 ⊻ (𝜓𝜒)) ↔ ¬ (𝜑 ↔ (𝜓𝜒)))
4 nbbn 388 . . 3 ((¬ 𝜑 ↔ (𝜓𝜒)) ↔ ¬ (𝜑 ↔ (𝜓𝜒)))
5 ifpdfbi 1066 . . 3 ((¬ 𝜑 ↔ (𝜓𝜒)) ↔ if-(¬ 𝜑, (𝜓𝜒), ¬ (𝜓𝜒)))
63, 4, 53bitr2i 302 . 2 ((𝜑 ⊻ (𝜓𝜒)) ↔ if-(¬ 𝜑, (𝜓𝜒), ¬ (𝜓𝜒)))
71, 2, 63bitr4i 306 1 (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  if-wif 1058  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-df3xor3  34880  wl-3xorbi  34883
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