Theorem cadifp 1626

Description: The value of the carry is, if the input carry is true, the disjunction, and if the input carry is false, the conjunction, of the other two inputs. (Contributed by BJ, 8-Oct-2019.)

Assertion

Ref	Expression
cadifp	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓)))

Proof of Theorem cadifp

Step	Hyp	Ref	Expression
1		cad1 1624	. 2 ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓)))
2		cad0 1625	. 2 ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
3	1, 2	casesifp 1083	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∨ wo 853  if-wif 1068  caddwcad 1613

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 208  df-an 397  df-or 854  df-ifp 1069  df-3or 1093  df-3an 1094  df-xor 1519  df-cad 1614

This theorem is referenced by:  wl-df-3mintru2  37853