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Theorem bdssexg 13161
Description: Bounded version of ssexg 4067. (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 3121 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21imbi1d 230 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 ∈ V) ↔ (𝐴𝐵𝐴 ∈ V)))
3 bdssexg.bd . . . 4 BOUNDED 𝐴
4 vex 2689 . . . 4 𝑥 ∈ V
53, 4bdssex 13159 . . 3 (𝐴𝑥𝐴 ∈ V)
62, 5vtoclg 2746 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 ∈ V))
76impcom 124 1 ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071  BOUNDED wbdc 13097
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13141
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-bdc 13098
This theorem is referenced by:  bdssexd  13162  bdrabexg  13163  bdunexb  13177
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