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Theorem euor 2029
 Description: Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
Hypothesis
Ref Expression
euor.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
euor ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Proof of Theorem euor
StepHypRef Expression
1 euor.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21hbn 1631 . . 3 𝜑 → ∀𝑥 ¬ 𝜑)
3 biorf 734 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3eubidh 2009 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
54biimpa 294 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1330  ∃!weu 2003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-eu 2006 This theorem is referenced by:  euorv  2030
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