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Theorem rabeqc 3658
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 3134 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abeq1 2944 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥𝐴))
3 rabeqc.1 . . . . 5 (𝑥𝐴𝜑)
43pm4.71i 562 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝜑))
54bicomi 226 . . 3 ((𝑥𝐴𝜑) ↔ 𝑥𝐴)
62, 5mpgbir 1800 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴
71, 6eqtri 2843 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  {cab 2798  {crab 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-rab 3134
This theorem is referenced by:  2clwwlk2  28112  numclwwlk3lem2lem  28147  fply1  30939  elnanelprv  32684
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