Theorem wl-df3xor2 35567

Description: Alternative definition of wl-df-3xor 35566, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1599. (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 1070	. 2 ⊢ (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
2		wl-df-3xor 35566	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)))
3		df-xor 1504	. . 3 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
4		nbbn 384	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒)))
5		ifpdfbi 1067	. . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
6	3, 4, 5	3bitr2i 298	. 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒)))
7	1, 2, 6	3bitr4i 302	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 205  if-wif 1059   ⊻ wxo 1503  haddwhad 1595

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 206  df-an 396  df-or 844  df-ifp 1060  df-xor 1504  df-tru 1542  df-had 1596

This theorem is referenced by:  wl-df3xor3  35568  wl-3xorbi  35571