Theorem hadifp 1606

Description: The value of the adder sum is, if the first input is true, the biconditionality, and if the first input is false, the exclusive disjunction, of the other two inputs. (Contributed by BJ, 11-Aug-2020.)

Assertion

Ref	Expression
hadifp	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)))

Proof of Theorem hadifp

Step	Hyp	Ref	Expression
1		had1 1604	. 2 ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))
2		had0 1605	. 2 ⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))
3	1, 2	casesifp 1071	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)))

Syntax hints:   ↔ 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:  wl-df-had  34748