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Theorem rspceaimv 2798
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
StepHypRef Expression
1 rspceaimv.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21imbi1d 230 . . 3  |-  ( x  =  A  ->  (
( ph  ->  ch )  <->  ( ps  ->  ch )
) )
32ralbidv 2438 . 2  |-  ( x  =  A  ->  ( A. y  e.  C  ( ph  ->  ch )  <->  A. y  e.  C  ( ps  ->  ch )
) )
43rspcev 2790 1  |-  ( ( A  e.  B  /\  A. y  e.  C  ( ps  ->  ch )
)  ->  E. x  e.  B  A. y  e.  C  ( ph  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689
This theorem is referenced by:  brimralrspcev  3991  reccn2ap  11110  metcnpi3  12716  elcncf1di  12765  mulcncflem  12789  limccnp2lem  12844
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