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Theorem rspcimdv 2764
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1  |-  ( ph  ->  A  e.  B )
rspcimdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
rspcimdv  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcimdv
StepHypRef Expression
1 df-ral 2398 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
2 rspcimdv.1 . . 3  |-  ( ph  ->  A  e.  B )
3 simpr 109 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
43eleq1d 2186 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
54biimprd 157 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
6 rspcimdv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
75, 6imim12d 74 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
82, 7spcimdv 2744 . . 3  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ( A  e.  B  ->  ch ) ) )
92, 8mpid 42 . 2  |-  ( ph  ->  ( A. x ( x  e.  B  ->  ps )  ->  ch )
)
101, 9syl5bi 151 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393
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-ral 2398  df-v 2662
This theorem is referenced by:  rspcdv  2766
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