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Theorem rabeqc 3517
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 3064 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abeq1 2876 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥𝐴))
3 rabeqc.1 . . . . 5 (𝑥𝐴𝜑)
43pm4.71i 555 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝜑))
54bicomi 215 . . 3 ((𝑥𝐴𝜑) ↔ 𝑥𝐴)
62, 5mpgbir 1894 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴
71, 6eqtri 2787 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  {crab 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-rab 3064
This theorem is referenced by:  2clwwlk2  27588  numclwwlk3lem2lem  27634
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