Theorem rspceaimv 2872

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 2494	. 2 |- ( x = A -> ( A. y e. C ( ph -> ch ) <-> A. y e. C ( ps -> ch ) ) )
4	3	rspcev 2864	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 1364   e. wcel 2164   A.wral 2472   E.wrex 2473

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762

This theorem is referenced by:  brimralrspcev  4088  reccn2ap  11456  metcnpi3  14685  elcncf1di  14734  mulcncflem  14761  limccnp2lem  14830