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Theorem bj-eunex 33151
Description: Remove dependency on ax-13 2411 from eunex 5004. (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 33149 . . . . 5 ¬ ∀𝑥 𝑥 = 𝑦
2 alim 1889 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
31, 2mtoi 190 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 2013 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
54adantl 468 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6 eu3v 2649 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
7 exnal 1905 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
85, 6, 73imtr4i 282 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1632  wex 1855  ∃!weu 2621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-nul 4936  ax-pow 4988
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-tru 1637  df-ex 1856  df-nf 1861  df-eu 2625  df-mo 2626
This theorem is referenced by: (None)
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