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Theorem pm13.18 3097
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 3078 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpd 230 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
32imp 407 1 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wne 3016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2814  df-ne 3017
This theorem is referenced by:  pm13.181  3099  iotan0  6339  frgrwopreglem5a  28018  4atexlemex4  37091  cncfiooicclem1  42056
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