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Theorem bdssexg 13029
Description: Bounded version of ssexg 4037. (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 3091 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21imbi1d 230 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 ∈ V) ↔ (𝐴𝐵𝐴 ∈ V)))
3 bdssexg.bd . . . 4 BOUNDED 𝐴
4 vex 2663 . . . 4 𝑥 ∈ V
53, 4bdssex 13027 . . 3 (𝐴𝑥𝐴 ∈ V)
62, 5vtoclg 2720 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 ∈ V))
76impcom 124 1 ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  wss 3041  BOUNDED wbdc 12965
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-bdc 12966
This theorem is referenced by:  bdssexd  13030  bdrabexg  13031  bdunexb  13045
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