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Theorem inex1g 3921
 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 3159 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2122 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 vex 2577 . . 3 𝑥 ∈ V
43inex1 3919 . 2 (𝑥𝐵) ∈ V
52, 4vtoclg 2630 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∈ wcel 1409  Vcvv 2574   ∩ cin 2944 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952 This theorem is referenced by:  onin  4151  dmresexg  4662  funimaexg  5011  offval  5747  offval3  5789
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