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Theorem euan 2686
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 2640 . . . 4 (∃!𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓))
2 moanim.1 . . . . 5 𝑥𝜑
3 simpl 486 . . . . 5 ((𝜑𝜓) → 𝜑)
42, 3exlimi 2216 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
51, 4syl 17 . . 3 (∃!𝑥(𝜑𝜓) → 𝜑)
6 ibar 532 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
72, 6eubid 2651 . . . 4 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
87biimprcd 253 . . 3 (∃!𝑥(𝜑𝜓) → (𝜑 → ∃!𝑥𝜓))
95, 8jcai 520 . 2 (∃!𝑥(𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
107biimpa 480 . 2 ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
119, 10impbii 212 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  wnf 1785  ∃!weu 2631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2601  df-eu 2632
This theorem is referenced by:  2eu7  2723  2eu8  2724
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