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Theorem bdinex1g 15050
Description: Bounded version of inex1g 4154. (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 3344 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2258 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 bdinex1g.bd . . 3 BOUNDED 𝐵
4 vex 2755 . . 3 𝑥 ∈ V
53, 4bdinex1 15048 . 2 (𝑥𝐵) ∈ V
62, 5vtoclg 2812 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  cin 3143  BOUNDED wbdc 14989
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bdsep 15033
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-bdc 14990
This theorem is referenced by: (None)
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