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Theorem euan 2619
Description: Introduction of a conjunct into unique existential 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 2575 . . . 4 (∃!𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓))
2 moanim.1 . . . . 5 𝑥𝜑
3 simpl 482 . . . . 5 ((𝜑𝜓) → 𝜑)
42, 3exlimi 2215 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
51, 4syl 17 . . 3 (∃!𝑥(𝜑𝜓) → 𝜑)
6 ibar 528 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
72, 6eubid 2585 . . . 4 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
87biimprcd 250 . . 3 (∃!𝑥(𝜑𝜓) → (𝜑 → ∃!𝑥𝜓))
95, 8jcai 516 . 2 (∃!𝑥(𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
107biimpa 476 . 2 ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
119, 10impbii 209 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wnf 1780  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567
This theorem is referenced by:  2eu7  2656  2eu8  2657
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