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Theorem bdssexg 9335
Description: Bounded version of ssexg 3887. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdssexg.bd BOUNDED A
Assertion
Ref Expression
bdssexg ((AB B 𝐶) → A V)

Proof of Theorem bdssexg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sseq2 2961 . . . 4 (x = B → (AxAB))
21imbi1d 220 . . 3 (x = B → ((AxA V) ↔ (ABA V)))
3 bdssexg.bd . . . 4 BOUNDED A
4 vex 2554 . . . 4 x V
53, 4bdssex 9333 . . 3 (AxA V)
62, 5vtoclg 2607 . 2 (B 𝐶 → (ABA V))
76impcom 116 1 ((AB B 𝐶) → A V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  BOUNDED wbdc 9275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9319
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-bdc 9276
This theorem is referenced by:  bdssexd  9336  bdrabexg  9337  bdunexb  9351
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