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Theorem pm13.18 2858
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 2609 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimprd 236 . . 3 (𝐴 = 𝐵 → (𝐵 = 𝐶𝐴 = 𝐶))
32necon3d 2798 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
43imp 443 1 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wne 2775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-cleq 2598  df-ne 2777
This theorem is referenced by:  pm13.181  2859  4atexlemex4  34176  cncfiooicclem1  38579
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