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Theorem spcdv 2684
Description: Rule of specialization, using implicit substitution. Analogous to rspcdv 2705. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
spcdv  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcdv
StepHypRef Expression
1 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
2 spcdv.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32biimpd 142 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
41, 3spcimdv 2683 1  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434
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-v 2604
This theorem is referenced by: (None)
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