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Theorem hadbi123i 1592
Description: Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
hadbii.1 (𝜑𝜓)
hadbii.2 (𝜒𝜃)
hadbii.3 (𝜏𝜂)
Assertion
Ref Expression
hadbi123i (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))

Proof of Theorem hadbi123i
StepHypRef Expression
1 hadbii.1 . . . 4 (𝜑𝜓)
21a1i 11 . . 3 (⊤ → (𝜑𝜓))
3 hadbii.2 . . . 4 (𝜒𝜃)
43a1i 11 . . 3 (⊤ → (𝜒𝜃))
5 hadbii.3 . . . 4 (𝜏𝜂)
65a1i 11 . . 3 (⊤ → (𝜏𝜂))
72, 4, 6hadbi123d 1591 . 2 (⊤ → (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂)))
87mptru 1540 1 (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wtru 1534  haddwhad 1589
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 1501  df-tru 1536  df-had 1590
This theorem is referenced by: (None)
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