Theorem rspceaimv 2919

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 2533	. 2 |- ( x = A -> ( A. y e. C ( ph -> ch ) <-> A. y e. C ( ps -> ch ) ) )
4	3	rspcev 2911	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 1398   e. wcel 2202   A.wral 2511   E.wrex 2512

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-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

This theorem is referenced by:  brimralrspcev  4153  reccn2ap  11953  mplsubgfilemm  14799  mplsubgfilemcl  14800  metcnpi3  15328  elcncf1di  15390  mulcncflem  15418  limccnp2lem  15487