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

Step	Hyp	Ref	Expression
1		breq2 3933	. . 3 |- ( x = B -> ( A R x <-> A R B ) )
2	1	ralbidv 2437	. 2 |- ( x = B -> ( A. y e. Y A R x <-> A. y e. Y A R B ) )
3	2	rspcev 2789	1 |- ( ( B e. X /\ A. y e. Y A R B ) -> E. x e. X A. y e. Y A R x )

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  7709  dedekindeulemub  12765  suplociccreex  12771  dedekindicclemub  12774