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Theorem euex 2030
 Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euex (∃!𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem euex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1507 . . 3 (𝜑 → ∀𝑦𝜑)
21eu1 2025 . 2 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
3 exsimpl 1597 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) → ∃𝑥𝜑)
42, 3sylbi 120 1 (∃!𝑥𝜑 → ∃𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469  [wsb 1736  ∃!weu 2000 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003 This theorem is referenced by:  eu2  2044  eu3h  2045  eu5  2047  exmoeudc  2063  eupickbi  2082  2eu2ex  2089  euxfrdc  2875  repizf  4053  eusvnf  4384  eusvnfb  4385  tz6.12c  5462  ndmfvg  5463  nfvres  5465  0fv  5467  eusvobj2  5771  fnoprabg  5883  txcn  12517
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