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Theorem rabeqc 3446
Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.)
Hypothesis
Ref Expression
rabeqc.1 (𝑥𝐴𝜑)
Assertion
Ref Expression
rabeqc {𝑥𝐴𝜑} = 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabeqc
StepHypRef Expression
1 rabeqc.1 . . . 4 (𝑥𝐴𝜑)
21adantl 481 . . 3 ((⊤ ∧ 𝑥𝐴) → 𝜑)
32rabeqcda 3445 . 2 (⊤ → {𝑥𝐴𝜑} = 𝐴)
43mptru 1544 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wtru 1538  wcel 2106  {crab 3433
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-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434
This theorem is referenced by:  bday0s  27888  2clwwlk2  30377  numclwwlk3lem2lem  30412  fply1  33564  elnanelprv  35414  ipolub0  48781  ipoglb0  48783
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