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Theorem rabeqc 3445
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 483 . . 3 ((⊤ ∧ 𝑥𝐴) → 𝜑)
32rabeqcda 3444 . 2 (⊤ → {𝑥𝐴𝜑} = 𝐴)
43mptru 1549 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wtru 1543  wcel 2107  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434
This theorem is referenced by:  bday0s  27329  2clwwlk2  29601  numclwwlk3lem2lem  29636  fply1  32637  elnanelprv  34420  ipolub0  47617  ipoglb0  47619
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