Theorem brralrspcev 3994

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 3941	. . 3 |- ( x = B -> ( A R x <-> A R B ) )
2	1	ralbidv 2438	. 2 |- ( x = B -> ( A. y e. Y A R x <-> A. y e. Y A R B ) )
3	2	rspcev 2793	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 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   class class class wbr 3937

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by:  axpre-suploclemres  7733  dedekindeulemub  12804  suplociccreex  12810  dedekindicclemub  12813