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Theorem eumo0 2050
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 2024 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 biimp 117 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43alimi 1448 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
54eximi 1593 . 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 1346  wex 1485  ∃!weu 2019
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-eu 2022
This theorem is referenced by:  eu2  2063  eu3h  2064
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