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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
StepHypRef Expression
1 rspceaimv.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21imbi1d 230 . . 3  |-  ( x  =  A  ->  (
( ph  ->  ch )  <->  ( ps  ->  ch )
) )
32ralbidv 2396 . 2  |-  ( x  =  A  ->  ( A. y  e.  C  ( ph  ->  ch )  <->  A. y  e.  C  ( ps  ->  ch )
) )
43rspcev 2744 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 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
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