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Theorem rabeqc 3615
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.)
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 df-rab 3073 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abeq1 2873 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥𝐴))
3 rabeqc.1 . . . . 5 (𝑥𝐴𝜑)
43pm4.71i 563 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝜑))
54bicomi 227 . . 3 ((𝑥𝐴𝜑) ↔ 𝑥𝐴)
62, 5mpgbir 1807 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴
71, 6eqtri 2767 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  {cab 2716  {crab 3068
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 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-rab 3073
This theorem is referenced by:  2clwwlk2  28462  numclwwlk3lem2lem  28497  fply1  31412  elnanelprv  33134  bday0s  33792  ipolub0  45996  ipoglb0  45998
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