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Theorem eumo0 2228
 Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1 yφ
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 yφ
21euf 2210 . 2 (∃!xφyx(φx = y))
3 bi1 178 . . . 4 ((φx = y) → (φx = y))
43alimi 1559 . . 3 (x(φx = y) → x(φx = y))
54eximi 1576 . 2 (yx(φx = y) → yx(φx = y))
62, 5sylbi 187 1 (∃!xφyx(φx = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by:  eu2  2229  mo2  2233
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