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Theorem bj-eumo0 31828
Description: Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2483 and mo2 2463. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)
Hypothesis
Ref Expression
bj-eumo0.1 𝑦𝜑
Assertion
Ref Expression
bj-eumo0 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-eumo0
StepHypRef Expression
1 bj-eumo0.1 . . 3 𝑦𝜑
21euf 2462 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 biimp 203 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43alimi 1729 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦))
54eximi 1751 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
62, 5sylbi 205 1 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  wnf 1698  ∃!weu 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2458
This theorem is referenced by: (None)
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