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Theorem euor2 1958
 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(𝜑𝜓) ↔ ∃!𝑥𝜓))

Proof of Theorem euor2
StepHypRef Expression
1 hbe1 1384 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbn 1544 . 2 (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
3 19.8a 1482 . . . 4 (𝜑 → ∃𝑥𝜑)
43con3i 562 . . 3 (¬ ∃𝑥𝜑 → ¬ 𝜑)
5 orel1 644 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
6 olc 632 . . . 4 (𝜓 → (𝜑𝜓))
75, 6impbid1 130 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
84, 7syl 14 . 2 (¬ ∃𝑥𝜑 → ((𝜑𝜓) ↔ 𝜓))
92, 8eubidh 1906 1 (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  ∃wex 1381  ∃!weu 1900 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-eu 1903 This theorem is referenced by:  reuun2  3220
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