Theorem brralrspcev 4147

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 4092	. . 3 |- ( x = B -> ( A R x <-> A R B ) )
2	1	ralbidv 2532	. 2 |- ( x = B -> ( A. y e. Y A R x <-> A. y e. Y A R B ) )
3	2	rspcev 2910	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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  axpre-suploclemres  8120  fiubm  11091  dedekindeulemub  15341  suplociccreex  15347  dedekindicclemub  15350