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Theorem inex1g 3952
Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
inex1g (𝐴𝑉 → (𝐴𝐵) ∈ V)

Proof of Theorem inex1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3183 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2153 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 vex 2618 . . 3 𝑥 ∈ V
43inex1 3950 . 2 (𝑥𝐵) ∈ V
52, 4vtoclg 2672 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  Vcvv 2615  cin 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994
This theorem is referenced by:  onin  4189  dmresexg  4705  funimaexg  5065  offval  5822  offval3  5864  ssenen  6521
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