Theorem wl-3xorbi 37474

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 37470	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
2		df-xor 1512	. 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
3		xor3 382	. . 3 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒)))
4		xnor 1513	. . . 4 ⊢ ((𝜓 ↔ 𝜒) ↔ ¬ (𝜓 ⊻ 𝜒))
5	4	bibi2i 337	. . 3 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ⊻ 𝜒)))
6	3, 5	bitr4i 278	. 2 ⊢ (¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
7	1, 2, 6	3bitri 297	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ⊻ wxo 1511  haddwhad 1593

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 207  df-an 396  df-or 849  df-ifp 1064  df-xor 1512  df-tru 1543  df-had 1594

This theorem is referenced by:  wl-3xorbi2  37475  wl-3xorrot  37478  wl-3xornot1  37481