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Theorem nexmo 2548
Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2093. (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 pm2.21 121 . . . . 5 𝜑 → (𝜑𝑥 = 𝑦))
21alimi 1774 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
32alrimiv 1886 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥(𝜑𝑥 = 𝑦))
4319.2d 1935 . 2 (∀𝑥 ¬ 𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 alnex 1744 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
65bicomi 216 . 2 (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
7 df-mo 2547 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
84, 6, 73imtr4i 284 1 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1505  wex 1742  ∃*wmo 2545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928
This theorem depends on definitions:  df-bi 199  df-ex 1743  df-mo 2547
This theorem is referenced by:  exmo  2550  moabs  2551  exmoeu  2600  moeuOLD  2617  moanimlem  2651  moexexvw  2661  2moswapv  2662  moexex  2669  mo2icl  3619  mosubopt  5256  dff3  6689  disjALTV0  35435
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