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Theorem rspcedvd 2769
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2767. (Contributed by AV, 27-Nov-2019.)
Hypotheses
Ref Expression
rspcedvd.1  |-  ( ph  ->  A  e.  B )
rspcedvd.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
rspcedvd.3  |-  ( ph  ->  ch )
Assertion
Ref Expression
rspcedvd  |-  ( ph  ->  E. x  e.  B  ps )
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcedvd
StepHypRef Expression
1 rspcedvd.3 . 2  |-  ( ph  ->  ch )
2 rspcedvd.1 . . 3  |-  ( ph  ->  A  e.  B )
3 rspcedvd.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
42, 3rspcedv 2767 . 2  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
51, 4mpd 13 1  |-  ( ph  ->  E. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by:  rspcime  2770  rspcedeq1vd  2772  rspcedeq2vd  2773  updjud  6935  modqmuladd  10107  modqmuladdnn0  10109  modfzo0difsn  10136  negfi  10967  divconjdvds  11474  2tp1odd  11508  dfgcd2  11629  qredeu  11705  pw2dvdslemn  11770  xmettx  12606
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