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Theorem eu3 2052
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eu3.1 𝑦𝜑
Assertion
Ref Expression
eu3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu3
StepHypRef Expression
1 eu3.1 . . 3 𝑦𝜑
21nfri 1499 . 2 (𝜑 → ∀𝑦𝜑)
32eu3h 2051 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333  wnf 1440  wex 1472  ∃!weu 2006
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009
This theorem is referenced by:  eqeu  2882  reu3  2902  eunex  4519
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