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Theorem bj-eunex 32495
Description: Remove dependency on ax-13 2245 from eunex 4829. (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 32493 . . . . 5 ¬ ∀𝑥 𝑥 = 𝑦
2 alim 1735 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
31, 2mtoi 190 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1855 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
54adantl 482 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6 eu3v 2497 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
7 exnal 1751 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
85, 6, 73imtr4i 281 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wex 1701  ∃!weu 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-nul 4759  ax-pow 4813
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2473  df-mo 2474
This theorem is referenced by: (None)
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