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Theorem hadnot 1538
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
StepHypRef Expression
1 notbi 309 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
21bibi1i 328 . 2 (((𝜑𝜓) ↔ ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
3 xor3 372 . . 3 (¬ ((𝜑𝜓) ↔ 𝜒) ↔ ((𝜑𝜓) ↔ ¬ 𝜒))
4 hadbi 1534 . . 3 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
53, 4xchnxbir 323 . 2 (¬ hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ ¬ 𝜒))
6 hadbi 1534 . 2 (hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜒))
72, 5, 63bitr4i 292 1 (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  haddwhad 1529
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 197  df-xor 1462  df-had 1530
This theorem is referenced by:  had0  1540
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