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Theorem eupick 2690
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 2625 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 mopick 2682 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
31, 2sylan 580 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1765  ∃*wmo 2576  ∃!weu 2613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614
This theorem is referenced by:  eupicka  2691  eupickb  2692  reupick  4213  reupick3  4214  eusv2nf  5194  reusv2lem3  5199  copsexg  5280  funssres  6275  oprabid  7054  txcn  21922  isch3  28705  bnj849  31809  iotasbc  40310
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