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Theorem wl-2xor 35165
 Description: In the recursive scheme "(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ) we set n = 1 to formally arrive at an expression for "2-xor". It is based on "1-xor", that is known to be equivalent to its only input (see wl-1xor 35164). (Contributed by Wolf Lammen, 11-May-2024.)
Assertion
Ref Expression
wl-2xor (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑𝜓))

Proof of Theorem wl-2xor
StepHypRef Expression
1 ifpdfbi 1067 . 2 ((¬ 𝜑𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓))
2 df-xor 1504 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
3 nbbn 389 . . 3 ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
42, 3bitr4i 281 . 2 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
5 ifpn 1070 . 2 (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓))
61, 4, 53bitr4ri 308 1 (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1059   ⊻ wxo 1503 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 401  df-or 846  df-ifp 1060  df-xor 1504 This theorem is referenced by: (None)
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