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Theorem reurab 3707
Description: Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
reurab.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
reurab (∃!𝑥 ∈ {𝑦𝐴𝜓}𝜒 ↔ ∃!𝑥𝐴 (𝜑𝜒))
Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem reurab
StepHypRef Expression
1 reurab.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
21bicomd 223 . . . . . . 7 (𝑥 = 𝑦 → (𝜓𝜑))
32equcoms 2019 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
43elrab 3692 . . . . 5 (𝑥 ∈ {𝑦𝐴𝜓} ↔ (𝑥𝐴𝜑))
54anbi1i 624 . . . 4 ((𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒) ↔ ((𝑥𝐴𝜑) ∧ 𝜒))
6 anass 468 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜑𝜒)))
75, 6bitri 275 . . 3 ((𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜑𝜒)))
87eubii 2585 . 2 (∃!𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒) ↔ ∃!𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
9 df-reu 3381 . 2 (∃!𝑥 ∈ {𝑦𝐴𝜓}𝜒 ↔ ∃!𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒))
10 df-reu 3381 . 2 (∃!𝑥𝐴 (𝜑𝜒) ↔ ∃!𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
118, 9, 103bitr4i 303 1 (∃!𝑥 ∈ {𝑦𝐴𝜓}𝜒 ↔ ∃!𝑥𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  ∃!weu 2568  ∃!wreu 3378  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-reu 3381  df-rab 3437  df-v 3482
This theorem is referenced by:  eqscut  27850
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