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Theorem nexmo 2530
Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2147. (Revised by Wolf Lammen, 16-Oct-2022.)
Assertion
Ref Expression
nexmo (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmo
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm2.21 123 . . . . 5 𝜑 → (𝜑𝑥 = 𝑦))
21alimi 1806 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
32alrimiv 1923 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥(𝜑𝑥 = 𝑦))
4319.2d 1974 . 2 (∀𝑥 ¬ 𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 alnex 1776 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
65bicomi 223 . 2 (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
7 df-mo 2529 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
84, 6, 73imtr4i 291 1 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  wex 1774  ∃*wmo 2527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964
This theorem depends on definitions:  df-bi 206  df-ex 1775  df-mo 2529
This theorem is referenced by:  exmo  2531  moabs  2532  exmoeu  2570  moanimlem  2607  moexexlem  2615  mo2icl  3708  mosubopt  5518  dff3  7116  disjALTV0  38454
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