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Theorem hadbi123d 1705
Description: Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
hadbid.1 (𝜑 → (𝜓𝜒))
hadbid.2 (𝜑 → (𝜃𝜏))
hadbid.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
hadbi123d (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))

Proof of Theorem hadbi123d
StepHypRef Expression
1 hadbid.1 . . . 4 (𝜑 → (𝜓𝜒))
2 hadbid.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2xorbi12d 1648 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 hadbid.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4xorbi12d 1648 . 2 (𝜑 → (((𝜓𝜃) ⊻ 𝜂) ↔ ((𝜒𝜏) ⊻ 𝜁)))
6 df-had 1704 . 2 (hadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ⊻ 𝜂))
7 df-had 1704 . 2 (hadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒𝜏) ⊻ 𝜁))
85, 6, 73bitr4g 306 1 (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wxo 1634  haddwhad 1703
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 199  df-xor 1635  df-had 1704
This theorem is referenced by:  hadbi123i  1706  sadfval  15509  sadval  15513
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