Theorem hadnot 1602

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

Assertion

Ref	Expression
hadnot	⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Proof of Theorem hadnot

Step	Hyp	Ref	Expression
1		notbi 321	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
2	1	bibi1i 341	. 2 ⊢ (((𝜑 ↔ 𝜓) ↔ ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
3		xor3 386	. . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
4		hadbi 1597	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
5	3, 4	xchnxbir 335	. 2 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ ¬ 𝜒))
6		hadbi 1597	. 2 ⊢ (hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
7	2, 5, 6	3bitr4i 305	1 ⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208  haddwhad 1592

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-xor 1502  df-had 1593

This theorem is referenced by:  had0  1604