Theorem rspceaimv 2751

Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)

Hypothesis

Ref	Expression
rspceaimv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rspceaimv	|- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) -> E. x e. B A. y e. C ( ph -> ch ) )

Distinct variable groups:   x, y, A   x, B   x, C   ps, x   ch, x

Allowed substitution hints:   ph( x, y)   ps( y)   ch( y)   B( y)   C( y)

Proof of Theorem rspceaimv

Step	Hyp	Ref	Expression
1		rspceaimv.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
2	1	imbi1d 230	. . 3 |- ( x = A -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
3	2	ralbidv 2396	. 2 |- ( x = A -> ( A. y e. C ( ph -> ch ) <-> A. y e. C ( ps -> ch ) ) )
4	3	rspcev 2744	1 |- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) -> E. x e. B A. y e. C ( ph -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   A.wral 2375   E.wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643

This theorem is referenced by:  reccn2ap  10921  metcnpi3  12441  elcncf1di  12479  mulcncflem  12502