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Theorem rabexg 4031
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 3148 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 ssexg 4027 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
31, 2mpan 418 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  {crab 2394  Vcvv 2657  wss 3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rab 2399  df-v 2659  df-in 3043  df-ss 3050
This theorem is referenced by:  rabex  4032  exmidsssnc  4086  exse  4218  frind  4234  elfvmptrab1  5469  mpoxopoveq  6091  diffitest  6734  epttop  12099  cldval  12108  neif  12150  neival  12152  cnfval  12203  cnpval  12206  ispsmet  12309  ismet  12330  isxmet  12331  blvalps  12374  blval  12375  cncfval  12542
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