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Theorem euor2 2272
 Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euor2 xφ → (∃!x(φ ψ) ↔ ∃!xψ))

Proof of Theorem euor2
StepHypRef Expression
1 nfe1 1732 . . 3 xxφ
21nfn 1793 . 2 x ¬ xφ
3 19.8a 1756 . . . 4 (φxφ)
43con3i 127 . . 3 xφ → ¬ φ)
5 orel1 371 . . . 4 φ → ((φ ψ) → ψ))
6 olc 373 . . . 4 (ψ → (φ ψ))
75, 6impbid1 194 . . 3 φ → ((φ ψ) ↔ ψ))
84, 7syl 15 . 2 xφ → ((φ ψ) ↔ ψ))
92, 8eubid 2211 1 xφ → (∃!x(φ ψ) ↔ ∃!xψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  ∃wex 1541  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-or 359  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by:  reuun2  3538
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