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Theorem bdinex1g 14893
Description: Bounded version of inex1g 4151. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdinex1g.bd BOUNDED 𝐵
Assertion
Ref Expression
bdinex1g (𝐴𝑉 → (𝐴𝐵) ∈ V)

Proof of Theorem bdinex1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3341 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2256 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 bdinex1g.bd . . 3 BOUNDED 𝐵
4 vex 2752 . . 3 𝑥 ∈ V
53, 4bdinex1 14891 . 2 (𝑥𝐵) ∈ V
62, 5vtoclg 2809 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  Vcvv 2749  cin 3140  BOUNDED wbdc 14832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bdsep 14876
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-bdc 14833
This theorem is referenced by: (None)
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