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Theorem euor 2696
Description: Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2697. (Contributed by NM, 21-Oct-2005.)
Hypothesis
Ref Expression
euor.nf 𝑥𝜑
Assertion
Ref Expression
euor ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Proof of Theorem euor
StepHypRef Expression
1 euor.nf . . . 4 𝑥𝜑
21nfn 1858 . . 3 𝑥 ¬ 𝜑
3 biorf 934 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3eubid 2672 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
54biimpa 480 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  wnf 1785  ∃!weu 2652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2622  df-eu 2653
This theorem is referenced by: (None)
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