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Theorem euan 2559
 Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
Hypothesis
Ref Expression
moanim.1 𝑥𝜑
Assertion
Ref Expression
euan (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Proof of Theorem euan
StepHypRef Expression
1 euex 2522 . . . 4 (∃!𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓))
2 moanim.1 . . . . 5 𝑥𝜑
3 simpl 472 . . . . 5 ((𝜑𝜓) → 𝜑)
42, 3exlimi 2124 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
51, 4syl 17 . . 3 (∃!𝑥(𝜑𝜓) → 𝜑)
6 ibar 524 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
72, 6eubid 2516 . . . 4 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
87biimprcd 240 . . 3 (∃!𝑥(𝜑𝜓) → (𝜑 → ∃!𝑥𝜓))
95, 8jcai 558 . 2 (∃!𝑥(𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
107biimpa 500 . 2 ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
119, 10impbii 199 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383  ∃wex 1744  Ⅎwnf 1748  ∃!weu 2498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502 This theorem is referenced by:  euanv  2563  2eu7  2588  2eu8  2589
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