Theorem wl-df3xor2 37429

Description: Alternative definition of wl-df-3xor 37428, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1596. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.)

Assertion

Ref	Expression
wl-df3xor2	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))

Proof of Theorem wl-df3xor2

Step	Hyp	Ref	Expression
1		ifpn 1073	. 2 ⊢ (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
2		wl-df-3xor 37428	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)))
3		df-xor 1511	. . 3 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
4		nbbn 383	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
5		ifpdfbi 1070	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
6	3, 4, 5	3bitr2i 299	. 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
7	1, 2, 6	3bitr4i 303	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 206  if-wif 1062   ⊻ wxo 1510  haddwhad 1592

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 848  df-ifp 1063  df-xor 1511  df-tru 1542  df-had 1593

This theorem is referenced by:  wl-df3xor3  37430  wl-3xorbi  37433