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Theorem bdssexg 13904
Description: Bounded version of ssexg 4126. (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 3171 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21imbi1d 230 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 ∈ V) ↔ (𝐴𝐵𝐴 ∈ V)))
3 bdssexg.bd . . . 4 BOUNDED 𝐴
4 vex 2733 . . . 4 𝑥 ∈ V
53, 4bdssex 13902 . . 3 (𝐴𝑥𝐴 ∈ V)
62, 5vtoclg 2790 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 ∈ V))
76impcom 124 1 ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  wss 3121  BOUNDED wbdc 13840
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bdsep 13884
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-bdc 13841
This theorem is referenced by:  bdssexd  13905  bdrabexg  13906  bdunexb  13920
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