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Theorem eu3 2021
 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 1482 . 2 (𝜑 → ∀𝑦𝜑)
32eu3h 2020 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1312  Ⅎwnf 1419  ∃wex 1451  ∃!weu 1975 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498 This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978 This theorem is referenced by:  eqeu  2825  reu3  2845  eunex  4444
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