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Theorem euor2 2006
Description: Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euor2 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2
StepHypRef Expression
1 hbe1 1429 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbn 1589 . 2 (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3 19.8a 1527 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 597 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 orel1 679 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
6 olc 667 . . . 4 (𝜓 → (𝜑𝜓))
75, 6impbid1 140 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
84, 7syl 14 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
92, 8eubidh 1954 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-eu 1951
This theorem is referenced by:  reuun2  3280
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