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Theorem bdinex1g 15837
Description: Bounded version of inex1g 4180. (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 3367 . . 3 (𝑥 = 𝐴 → (𝑥𝐵) = (𝐴𝐵))
21eleq1d 2274 . 2 (𝑥 = 𝐴 → ((𝑥𝐵) ∈ V ↔ (𝐴𝐵) ∈ V))
3 bdinex1g.bd . . 3 BOUNDED 𝐵
4 vex 2775 . . 3 𝑥 ∈ V
53, 4bdinex1 15835 . 2 (𝑥𝐵) ∈ V
62, 5vtoclg 2833 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1373  wcel 2176  Vcvv 2772  cin 3165  BOUNDED wbdc 15776
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bdsep 15820
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-bdc 15777
This theorem is referenced by: (None)
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