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Theorem eupick 2428
Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
Assertion
Ref Expression
eupick ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem eupick
StepHypRef Expression
1 eumo 2391 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 mopick 2427 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
31, 2sylan 486 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1694  ∃!weu 2362  ∃*wmo 2363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699  df-eu 2366  df-mo 2367
This theorem is referenced by:  eupicka  2429  eupickb  2430  reupick  3773  reupick3  3774  eusv2nf  4689  reusv2lem3  4696  copsexg  4780  funssres  5729  oprabid  6452  txcn  21140  isch3  27271  bnj849  30095  iotasbc  37524
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