Theorem wl-3xortru 37903

Description: If the first input is true, then triple xor is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.)

Assertion

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

Proof of Theorem wl-3xortru

Step	Hyp	Ref	Expression
1		wl-df-3xor 37900	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)))
2		ifptru 1083	. 2 ⊢ (𝜑 → (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜓 ⊻ 𝜒)))
3	1, 2	bitrid 285	1 ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ ¬ (𝜓 ⊻ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  if-wif 1071   ⊻ wxo 1521  haddwhad 1603

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 209  df-an 399  df-or 857  df-ifp 1072  df-xor 1522  df-tru 1553  df-had 1604

This theorem is referenced by: (None)