Theorem cadnot 1612

Description: The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Assertion

Ref	Expression
cadnot	⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Proof of Theorem cadnot

Step	Hyp	Ref	Expression
1		ianor 978	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2		ianor 978	. . 3 ⊢ (¬ (𝜑 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜒))
3		ianor 978	. . 3 ⊢ (¬ (𝜓 ∧ 𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
4	1, 2, 3	3anbi123i 1151	. 2 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ ¬ (𝜑 ∧ 𝜒) ∧ ¬ (𝜓 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
5		3ioran 1102	. . 3 ⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ¬ (𝜑 ∧ 𝜒) ∧ ¬ (𝜓 ∧ 𝜒)))
6		cador 1605	. . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
7	5, 6	xchnxbir 335	. 2 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ (¬ (𝜑 ∧ 𝜓) ∧ ¬ (𝜑 ∧ 𝜒) ∧ ¬ (𝜓 ∧ 𝜒)))
8		cadan 1606	. 2 ⊢ (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
9	4, 7, 8	3bitr4i 305	1 ⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083  caddwcad 1603

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-3or 1084  df-3an 1085  df-xor 1501  df-cad 1604

This theorem is referenced by: (None)