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Theorem euanv 2708
Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
Assertion
Ref Expression
euanv (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euanv
StepHypRef Expression
1 euex 2660 . . . 4 (∃!𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓))
2 simpl 483 . . . . 5 ((𝜑𝜓) → 𝜑)
32exlimiv 1924 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
41, 3syl 17 . . 3 (∃!𝑥(𝜑𝜓) → 𝜑)
5 ibar 529 . . . . 5 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
65eubidv 2670 . . . 4 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
76biimprcd 251 . . 3 (∃!𝑥(𝜑𝜓) → (𝜑 → ∃!𝑥𝜓))
84, 7jcai 517 . 2 (∃!𝑥(𝜑𝜓) → (𝜑 ∧ ∃!𝑥𝜓))
96biimpa 477 . 2 ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
108, 9impbii 210 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1773  ∃!weu 2651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-mo 2620  df-eu 2652
This theorem is referenced by:  eueq2  3705  2reu5lem1  3750  fsn  6895  dfac5lem5  9547
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