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Theorem spcimdv 2969
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
spcimdv  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimdv
StepHypRef Expression
1 spcimdv.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
21ex 424 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  ->  ch ) ) )
32alrimiv 1638 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps  ->  ch ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1626 . . 3  |-  F/ x ch
6 nfcv 2516 . . 3  |-  F/_ x A
75, 6spcimgft 2963 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ch ) )  -> 
( A  e.  B  ->  ( A. x ps 
->  ch ) ) )
83, 4, 7sylc 58 1  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717
This theorem is referenced by:  spcdv  2970  spcimedv  2971  rspcimdv  2989  mrieqv2d  13784  mreexexlemd  13789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894
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