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Theorem pm13.14 41700
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 3705 . . . 4 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21biimprcd 253 . . 3 ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴𝜑))
32necon3bd 2954 . 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 1543  wne 2940  [wsbc 3694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-sbc 3695
This theorem is referenced by: (None)
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