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Theorem rabexg 4074
Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
Assertion
Ref Expression
rabexg (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabexg
StepHypRef Expression
1 ssrab2 3182 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 ssexg 4070 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
31, 2mpan 420 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  {crab 2420  Vcvv 2686  wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084
This theorem is referenced by:  rabex  4075  exmidsssnc  4129  exse  4261  frind  4277  elfvmptrab1  5518  mpoxopoveq  6140  diffitest  6784  cc4f  7096  omctfn  11979  epttop  12285  cldval  12294  neif  12336  neival  12338  cnfval  12389  cnovex  12391  cnpval  12393  hmeofn  12497  hmeofvalg  12498  ispsmet  12518  ismet  12539  isxmet  12540  blvalps  12583  blval  12584  cncfval  12754
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