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Theorem rabeqc 3511
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 3069 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 abeq1 2881 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ ∀𝑥((𝑥𝐴𝜑) ↔ 𝑥𝐴))
3 rabeqc.1 . . . . 5 (𝑥𝐴𝜑)
43pm4.71i 541 . . . 4 (𝑥𝐴 ↔ (𝑥𝐴𝜑))
54bicomi 214 . . 3 ((𝑥𝐴𝜑) ↔ 𝑥𝐴)
62, 5mpgbir 1873 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = 𝐴
71, 6eqtri 2792 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  {cab 2756  {crab 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-rab 3069
This theorem is referenced by:  2clwwlk2  27530  numclwwlk3lemlem  27576
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