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Theorem rspcdv 2705
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1  |-  ( ph  ->  A  e.  B )
rspcdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rspcdv  |-  ( 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 rspcdv
StepHypRef Expression
1 rspcdv.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcdv.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32biimpd 142 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
41, 3rspcimdv 2703 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604
This theorem is referenced by:  rspcdva  2708  ralxfrd  4214  prloc  6732  zindd  8535
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