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Theorem rabexg 4132
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 3232 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 ssexg 4128 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
31, 2mpan 422 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  {crab 2452  Vcvv 2730  wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134
This theorem is referenced by:  rabex  4133  exmidsssnc  4189  exse  4321  frind  4337  elfvmptrab1  5590  mpoxopoveq  6219  diffitest  6865  supex2g  7010  cc4f  7231  omctfn  12398  ismhm  12685  issubm  12695  epttop  12884  cldval  12893  neif  12935  neival  12937  cnfval  12988  cnovex  12990  cnpval  12992  hmeofn  13096  hmeofvalg  13097  ispsmet  13117  ismet  13138  isxmet  13139  blvalps  13182  blval  13183  cncfval  13353
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