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Theorem euor2 1940
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 hbe1 1365 . . 3 (xφxxφ)
21hbn 1526 . 2 xφx ¬ xφ)
3 19.8a 1464 . . . 4 (φxφ)
43con3i 549 . . 3 xφ → ¬ φ)
5 orel1 631 . . . 4 φ → ((φ ψ) → ψ))
6 olc 619 . . . 4 (ψ → (φ ψ))
75, 6impbid1 130 . . 3 φ → ((φ ψ) ↔ ψ))
84, 7syl 14 . 2 xφ → ((φ ψ) ↔ ψ))
92, 8eubidh 1888 1 xφ → (∃!x(φ ψ) ↔ ∃!xψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 616  wex 1362  ∃!weu 1882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-eu 1885
This theorem is referenced by:  reuun2  3197
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