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Theorem bdinex1g 13476
Description: Bounded version of inex1g 4100. (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 3301 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2226 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 bdinex1g.bd . . 3 BOUNDED 𝐵
4 vex 2715 . . 3 𝑥 ∈ V
53, 4bdinex1 13474 . 2 (𝑥𝐵) ∈ V
62, 5vtoclg 2772 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Vcvv 2712  cin 3101  BOUNDED wbdc 13415
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bdsep 13459
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-bdc 13416
This theorem is referenced by: (None)
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