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Theorem brralrspcev 3986
Description: Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
Assertion
Ref Expression
brralrspcev  |-  ( ( B  e.  X  /\  A. y  e.  Y  A R B )  ->  E. x  e.  X  A. y  e.  Y  A R x )
Distinct variable groups:    x, A    x, B, y    x, R    x, X    x, Y
Allowed substitution hints:    A( y)    R( y)    X( y)    Y( y)

Proof of Theorem brralrspcev
StepHypRef Expression
1 breq2 3933 . . 3  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21ralbidv 2437 . 2  |-  ( x  =  B  ->  ( A. y  e.  Y  A R x  <->  A. y  e.  Y  A R B ) )
32rspcev 2789 1  |-  ( ( B  e.  X  /\  A. y  e.  Y  A R B )  ->  E. x  e.  X  A. y  e.  Y  A R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929
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
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  axpre-suploclemres  7716  dedekindeulemub  12774  suplociccreex  12780  dedekindicclemub  12783
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