Theorem wl-had1 34762

Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)

Assertion

Ref	Expression
wl-had1	⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))

Proof of Theorem wl-had1

Step	Hyp	Ref	Expression
1		wl-df-had 34761	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)))
2		ifptru 1070	. 2 ⊢ (𝜑 → (if-(𝜑, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)) ↔ (𝜓 ↔ 𝜒)))
3	1, 2	syl5bb 285	1 ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208  if-wif 1057   ⊻ wxo 1501  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 209  df-an 399  df-or 844  df-ifp 1058  df-xor 1502  df-had 1594

This theorem is referenced by: (None)