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Theorem pm13.14 41284
 Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.14 (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)

Proof of Theorem pm13.14
StepHypRef Expression
1 sbceq1a 3733 . . . 4 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21biimprcd 253 . . 3 ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴𝜑))
32necon3bd 3001 . 2 ([𝐴 / 𝑥]𝜑 → (¬ 𝜑𝑥𝐴))
43imp 410 1 (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 2987  [wsbc 3722 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-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-sbc 3723 This theorem is referenced by: (None)
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