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Theorem riotaclb 7230
Description: Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
Assertion
Ref Expression
riotaclb (¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaclb
StepHypRef Expression
1 riotacl 7206 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
2 riotaund 7228 . . . . . 6 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
32eleq1d 2823 . . . . 5 (¬ ∃!𝑥𝐴 𝜑 → ((𝑥𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴))
43notbid 321 . . . 4 (¬ ∃!𝑥𝐴 𝜑 → (¬ (𝑥𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
54biimprcd 253 . . 3 (¬ ∅ ∈ 𝐴 → (¬ ∃!𝑥𝐴 𝜑 → ¬ (𝑥𝐴 𝜑) ∈ 𝐴))
65con4d 115 . 2 (¬ ∅ ∈ 𝐴 → ((𝑥𝐴 𝜑) ∈ 𝐴 → ∃!𝑥𝐴 𝜑))
71, 6impbid2 229 1 (¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wcel 2111  ∃!wreu 3064  c0 4251  crio 7187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3422  df-sbc 3709  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-sn 4556  df-pr 4558  df-uni 4834  df-iota 6355  df-riota 7188
This theorem is referenced by: (None)
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