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Theorem wl-df-3xor 35036
Assertion
Ref Expression
wl-df-3xor (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒)))

Proof of Theorem wl-df-3xor
StepHypRef Expression
1 hadifp 1607 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓𝜒), (𝜓𝜒)))
2 xnor 1504 . . . . 5 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32a1i 11 . . . 4 (⊤ → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
4 biidd 265 . . . 4 (⊤ → ((𝜓𝜒) ↔ (𝜓𝜒)))
53, 4ifpbi23d 1077 . . 3 (⊤ → (if-(𝜑, (𝜓𝜒), (𝜓𝜒)) ↔ if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒))))
65mptru 1545 . 2 (if-(𝜑, (𝜓𝜒), (𝜓𝜒)) ↔ if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒)))
71, 6bitri 278 1 (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓𝜒), (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1058   ⊻ wxo 1502  ⊤wtru 1539  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-df3xor2  35037  wl-3xortru  35039  wl-3xorfal  35040
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