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Theorem eumo0 1928
Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1 (φyφ)
Assertion
Ref Expression
eumo0 (∃!xφyx(φx = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eumo0
StepHypRef Expression
1 eumo0.1 . . 3 (φyφ)
21euf 1902 . 2 (∃!xφyx(φx = y))
3 bi1 111 . . . 4 ((φx = y) → (φx = y))
43alimi 1341 . . 3 (x(φx = y) → x(φx = y))
54eximi 1488 . 2 (yx(φx = y) → yx(φx = y))
62, 5sylbi 114 1 (∃!xφyx(φx = y))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-eu 1900
This theorem is referenced by:  eu2  1941  eu3h  1942
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