Theorem wl-3xorbi 38147

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 ↔ (𝜓 ↔ 𝜒)))

Proof of Theorem wl-3xorbi

Step	Hyp	Ref	Expression
1		wl-df3xor2 38143	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
2		df-xor 1541	. 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
3		xor3 385	. . 3 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒)))
4		xnor 1542	. . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ ¬ (𝜓 ⊻ 𝜒))
5	4	bibi2i 340	. . 3 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒)))
6	3, 5	bitr4i 281	. 2 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
7	1, 2, 6	3bitri 300	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1540  haddwhad 1622

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1078  df-xor 1541  df-tru 1572  df-had 1623

This theorem is used by:  wl-3xorbi2  38148  wl-3xorrot  38151  wl-3xornot1  38154