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Theorem pm13.18 2301
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 2062 . . . 4 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21biimprd 151 . . 3 (𝐴 = 𝐵 → (𝐵 = 𝐶𝐴 = 𝐶))
32necon3d 2264 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
43imp 119 1 ((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221
This theorem is referenced by:  pm13.181  2302
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