Theorem wl-3xorfal 35570

Description: If the first input is false, then triple xor is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 29-Apr-2024.)

Assertion

Ref	Expression
wl-3xorfal	⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))

Proof of Theorem wl-3xorfal

Step	Hyp	Ref	Expression
1		wl-df-3xor 35566	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)))
2		ifpfal 1073	. 2 ⊢ (¬ 𝜑 → (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ (𝜓 ⊻ 𝜒)))
3	1, 2	syl5bb 282	1 ⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ 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: (None)