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Theorem pm13.18 3068
 Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 14-May-2023.)
Assertion
Ref Expression
pm13.18 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Proof of Theorem pm13.18
StepHypRef Expression
1 neeq1 3049 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpd 232 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
32imp 410 1 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 2987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-ne 2988 This theorem is referenced by:  pm13.181  3069  iotan0  6322  frgrwopreglem5a  28140  4atexlemex4  37520  cncfiooicclem1  42703
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