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Theorem bj-eunex 33375
 Description: Remove dependency on ax-13 2334 from eunex 5101. (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 33373 . . . 4 ¬ ∀𝑥 𝑥 = 𝑦
2 albi 1862 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦))
31, 2mtbiri 319 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1973 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5 eu6 2592 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 exnal 1870 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
74, 5, 63imtr4i 284 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  ∃!weu 2586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-nul 5025  ax-pow 5077 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551  df-eu 2587 This theorem is referenced by: (None)
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