Theorem cadifp 1619

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 1617	. 2 ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓)))
2		cad0 1618	. 2 ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
3	1, 2	casesifp 1077	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847  if-wif 1062  caddwcad 1606

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 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-xor 1512  df-cad 1607

This theorem is referenced by:  wl-df-3mintru2  37507