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Theorem nexmo 2599
 Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2158. (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 1813 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
32alrimiv 1928 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥(𝜑𝑥 = 𝑦))
4319.2d 1982 . 2 (∀𝑥 ¬ 𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 alnex 1783 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
65bicomi 227 . 2 (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
7 df-mo 2598 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
84, 6, 73imtr4i 295 1 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781  ∃*wmo 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-mo 2598 This theorem is referenced by:  exmo  2600  moabs  2601  exmoeu  2641  moanimlem  2680  moexexlem  2688  mo2icl  3653  mosubopt  5365  dff3  6843  disjALTV0  36160
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