Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cadbi123i Structured version   Visualization version   GIF version

Theorem cadbi123i 1590
 Description: Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
cadbii.1 (𝜑𝜓)
cadbii.2 (𝜒𝜃)
cadbii.3 (𝜏𝜂)
Assertion
Ref Expression
cadbi123i (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))

Proof of Theorem cadbi123i
StepHypRef Expression
1 cadbii.1 . . . 4 (𝜑𝜓)
21a1i 11 . . 3 (⊤ → (𝜑𝜓))
3 cadbii.2 . . . 4 (𝜒𝜃)
43a1i 11 . . 3 (⊤ → (𝜒𝜃))
5 cadbii.3 . . . 4 (𝜏𝜂)
65a1i 11 . . 3 (⊤ → (𝜏𝜂))
72, 4, 6cadbi123d 1589 . 2 (⊤ → (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂)))
87trud 1533 1 (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ⊤wtru 1524  caddwcad 1585 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-or 384  df-an 385  df-xor 1505  df-tru 1526  df-cad 1586 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator