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Theorem eumo0 2006
 Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
eumo0 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eumo0
StepHypRef Expression
1 eumo0.1 . . 3 (𝜑 → ∀𝑦𝜑)
21euf 1980 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 bi1 117 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43alimi 1414 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
54eximi 1562 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
62, 5sylbi 120 1 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1312  ∃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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498 This theorem depends on definitions:  df-bi 116  df-eu 1978 This theorem is referenced by:  eu2  2019  eu3h  2020
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