MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabeqc Structured version   Visualization version   GIF version

Theorem rabeqc 3449
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 481 . . 3 ((⊤ ∧ 𝑥𝐴) → 𝜑)
32rabeqcda 3448 . 2 (⊤ → {𝑥𝐴𝜑} = 𝐴)
43mptru 1547 1 {𝑥𝐴𝜑} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wtru 1541  wcel 2108  {crab 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437
This theorem is referenced by:  bday0s  27873  2clwwlk2  30367  numclwwlk3lem2lem  30402  fply1  33584  elnanelprv  35434  ipolub0  48881  ipoglb0  48883
  Copyright terms: Public domain W3C validator