MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm13.18 Structured version   Visualization version   GIF version

Theorem pm13.18 2874
Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.18 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Proof of Theorem pm13.18
StepHypRef Expression
1 eqeq1 2625 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimprd 238 . . 3 (𝐴 = 𝐵 → (𝐵 = 𝐶𝐴 = 𝐶))
32necon3d 2814 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
43imp 445 1 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wne 2793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-cleq 2614  df-ne 2794
This theorem is referenced by:  pm13.181  2875  4atexlemex4  35185  cncfiooicclem1  39875
  Copyright terms: Public domain W3C validator