Theorem cadifp 1622

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 1619	. 2 ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓)))
2		cad0 1620	. 2 ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
3	1, 2	casesifp 1076	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060  caddwcad 1608

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 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-xor 1507  df-cad 1609

This theorem is referenced by:  wl-df-3mintru2  35655