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Theorem hadbi 1707
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 1634 . 2 (((𝜑𝜓) ⊻ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
2 df-had 1703 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
3 xnor 1635 . . . 4 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
43bibi1i 329 . . 3 (((𝜑𝜓) ↔ 𝜒) ↔ (¬ (𝜑𝜓) ↔ 𝜒))
5 nbbn 374 . . 3 ((¬ (𝜑𝜓) ↔ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
64, 5bitri 266 . 2 (((𝜑𝜓) ↔ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
71, 2, 63bitr4i 294 1 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wxo 1633  haddwhad 1702
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 198  df-xor 1634  df-had 1703
This theorem is referenced by:  hadnot  1711  had1  1712
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