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Theorem eu3 1924
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 1389 . 2 (φyφ)
32eu3h 1923 1 (∃!xφ ↔ (xφ yx(φx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224  wnf 1325  wex 1358  ∃!weu 1878
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-eu 1881
This theorem is referenced by:  eqeu  2684  reu3  2704  eunex  4219
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