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Theorem bj-eunex 31987
 Description: Remove dependency on ax-13 2234 from eunex 4785. (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-eunex (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem bj-eunex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-dtru 31985 . . . . 5 ¬ ∀𝑥 𝑥 = 𝑦
2 alim 1729 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
31, 2mtoi 189 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1845 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
54adantl 481 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6 eu3v 2486 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
7 exnal 1744 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
85, 6, 73imtr4i 280 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by: (None)
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