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Theorem pm13.14 37431
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 3408 . . . 4 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21biimprcd 238 . . 3 ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴𝜑))
32necon3bd 2791 . 2 ([𝐴 / 𝑥]𝜑 → (¬ 𝜑𝑥𝐴))
43imp 443 1 (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wne 2775  [wsbc 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-ne 2777  df-sbc 3398
This theorem is referenced by: (None)
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