Theorem brralrspcev 4152

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 4097	. . 3 |- ( x = B -> ( A R x <-> A R B ) )
2	1	ralbidv 2533	. 2 |- ( x = B -> ( A. y e. Y A R x <-> A. y e. Y A R B ) )
3	2	rspcev 2911	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 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  axpre-suploclemres  8181  fiubm  11155  dedekindeulemub  15429  suplociccreex  15435  dedekindicclemub  15438