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Theorem bdssexg 11795
Description: Bounded version of ssexg 3978. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdssexg.bd BOUNDED 𝐴
Assertion
Ref Expression
bdssexg ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Proof of Theorem bdssexg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3048 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21imbi1d 229 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 ∈ V) ↔ (𝐴𝐵𝐴 ∈ V)))
3 bdssexg.bd . . . 4 BOUNDED 𝐴
4 vex 2622 . . . 4 𝑥 ∈ V
53, 4bdssex 11793 . . 3 (𝐴𝑥𝐴 ∈ V)
62, 5vtoclg 2679 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 ∈ V))
76impcom 123 1 ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  wss 2999  BOUNDED wbdc 11731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-bdc 11732
This theorem is referenced by:  bdssexd  11796  bdrabexg  11797  bdunexb  11811
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