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Theorem brimralrspcev 4061
Description: Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
Assertion
Ref Expression
brimralrspcev  |-  ( ( B  e.  X  /\  A. y  e.  Y  ( ( ph  /\  A R B )  ->  ps ) )  ->  E. x  e.  X  A. y  e.  Y  ( ( ph  /\  A R x )  ->  ps )
)
Distinct variable groups:    x, A    x, B, y    x, R    x, X    x, Y    ph, x    ps, x
Allowed substitution hints:    ph( y)    ps( y)    A( y)    R( y)    X( y)    Y( y)

Proof of Theorem brimralrspcev
StepHypRef Expression
1 breq2 4006 . . 3  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21anbi2d 464 . 2  |-  ( x  =  B  ->  (
( ph  /\  A R x )  <->  ( ph  /\  A R B ) ) )
32rspceaimv 2849 1  |-  ( ( B  e.  X  /\  A. y  e.  Y  ( ( ph  /\  A R B )  ->  ps ) )  ->  E. x  e.  X  A. y  e.  Y  ( ( ph  /\  A R x )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003
This theorem is referenced by:  dveflem  14058
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