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Theorem euor 2500
 Description: Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
Hypothesis
Ref Expression
euor.1 𝑥𝜑
Assertion
Ref Expression
euor ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Proof of Theorem euor
StepHypRef Expression
1 euor.1 . . . 4 𝑥𝜑
21nfn 1768 . . 3 𝑥 ¬ 𝜑
3 biorf 419 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3eubid 2476 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
54biimpa 500 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  Ⅎwnf 1699  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-eu 2462 This theorem is referenced by:  euorv  2501
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