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Theorem bdssex 9333
 Description: Bounded version of ssex 3885. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdssex.bd BOUNDED A
bdssex.1 B V
Assertion
Ref Expression
bdssex (ABA V)

Proof of Theorem bdssex
StepHypRef Expression
1 df-ss 2925 . 2 (AB ↔ (AB) = A)
2 bdssex.bd . . . 4 BOUNDED A
3 bdssex.1 . . . 4 B V
42, 3bdinex2 9331 . . 3 (AB) V
5 eleq1 2097 . . 3 ((AB) = A → ((AB) V ↔ A V))
64, 5mpbii 136 . 2 ((AB) = AA V)
71, 6sylbi 114 1 (ABA V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ⊆ 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:  bdssexi  9334  bdssexg  9335  bdfind  9380
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