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Theorem wl-1xor 35265
Description: In the recursive scheme

"(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" )

we set n = 0 to formally arrive at an expression for "1-xor". The base case "0-xor" is replaced with , as a sequence of 0 inputs never has an odd number being part of it. (Contributed by Wolf Lammen, 11-May-2024.)

Assertion
Ref Expression
wl-1xor (if-(𝜓, ¬ ⊥, ⊥) ↔ 𝜓)

Proof of Theorem wl-1xor
StepHypRef Expression
1 tbtru 1550 . . . . 5 (𝜓 ↔ (𝜓 ↔ ⊤))
21biimpi 219 . . . 4 (𝜓 → (𝜓 ↔ ⊤))
3 notfal 1570 . . . 4 (¬ ⊥ ↔ ⊤)
42, 3bitr4di 292 . . 3 (𝜓 → (𝜓 ↔ ¬ ⊥))
5 nbfal 1557 . . . 4 𝜓 ↔ (𝜓 ↔ ⊥))
65biimpi 219 . . 3 𝜓 → (𝜓 ↔ ⊥))
74, 6casesifp 1078 . 2 (𝜓 ↔ if-(𝜓, ¬ ⊥, ⊥))
87bicomi 227 1 (if-(𝜓, ¬ ⊥, ⊥) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  if-wif 1062  wtru 1543  wfal 1554
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 847  df-ifp 1063  df-tru 1545  df-fal 1555
This theorem is referenced by: (None)
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