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Theorem had1 1539
Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
Assertion
Ref Expression
had1 (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)))

Proof of Theorem had1
StepHypRef Expression
1 hadrot 1537 . . . 4 (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
2 hadbi 1534 . . . 4 (hadd(𝜓, 𝜒, 𝜑) ↔ ((𝜓𝜒) ↔ 𝜑))
31, 2bitri 264 . . 3 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓𝜒) ↔ 𝜑))
4 biass 374 . . 3 (((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)) ↔ 𝜑) ↔ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓𝜒) ↔ 𝜑)))
53, 4mpbir 221 . 2 ((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)) ↔ 𝜑)
65biimpri 218 1 (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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  hadifp  1541  sadadd2lem2  15115
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