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Theorem euex 2056
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 1526 . . 3 (𝜑 → ∀𝑦𝜑)
21eu1 2051 . 2 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
3 exsimpl 1617 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) → ∃𝑥𝜑)
42, 3sylbi 121 1 (∃!𝑥𝜑 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351  wex 1492  [wsb 1762  ∃!weu 2026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029
This theorem is referenced by:  eu2  2070  eu3h  2071  eu5  2073  exmoeudc  2089  eupickbi  2108  2eu2ex  2115  euxfrdc  2923  repizf  4116  eusvnf  4450  eusvnfb  4451  tz6.12c  5541  ndmfvg  5542  nfvres  5544  0fv  5546  eusvobj2  5855  fnoprabg  5970  0g0  12684  txcn  13435
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