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 Description: The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadbi (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))

Proof of Theorem hadbi
StepHypRef Expression
1 df-xor 1503 . 2 (((𝜑𝜓) ⊻ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
2 df-had 1595 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
3 xnor 1504 . . . 4 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
43bibi1i 342 . . 3 (((𝜑𝜓) ↔ 𝜒) ↔ (¬ (𝜑𝜓) ↔ 𝜒))
5 nbbn 388 . . 3 ((¬ (𝜑𝜓) ↔ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
64, 5bitri 278 . 2 (((𝜑𝜓) ↔ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
71, 2, 63bitr4i 306 1 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1502  haddwhad 1594 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 210  df-xor 1503  df-had 1595 This theorem is referenced by:  hadcoma  1600  hadnot  1604  had1  1605
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