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Theorem eumo0 1947
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 1921 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 bi1 115 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43alimi 1360 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
54eximi 1507 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
62, 5sylbi 118 1 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-eu 1919
This theorem is referenced by:  eu2  1960  eu3h  1961
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