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Theorem eupick 1995
 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 1948 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 mopick 1994 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
31, 2sylan 271 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101  ∃wex 1397  ∃!weu 1916  ∃*wmo 1917 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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920 This theorem is referenced by:  eupicka  1996  eupickb  1997  reupick  3249  reupick3  3250  copsexg  4009  eusv2nf  4216  funssres  4970  oprabid  5565
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