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Theorem eu3 1943
 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 1409 . 2 (φyφ)
32eu3h 1942 1 (∃!xφ ↔ (xφ yx(φx = y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378  ∃!weu 1897 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900 This theorem is referenced by:  eqeu  2705  reu3  2725  eunex  4239
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