Theorem rspceaimv 2851

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 231	. . 3 |- ( x = A -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
3	2	ralbidv 2477	. 2 |- ( x = A -> ( A. y e. C ( ph -> ch ) <-> A. y e. C ( ps -> ch ) ) )
4	3	rspcev 2843	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 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741

This theorem is referenced by:  brimralrspcev  4064  reccn2ap  11323  metcnpi3  14056  elcncf1di  14105  mulcncflem  14129  limccnp2lem  14184