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Theorem spcimdv 3153
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 434 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  ->  ch ) ) )
32alrimiv 1686 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps  ->  ch ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1674 . . 3  |-  F/ x ch
6 nfcv 2613 . . 3  |-  F/_ x A
75, 6spcimgft 3147 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ch ) )  -> 
( A  e.  B  ->  ( A. x ps 
->  ch ) ) )
83, 4, 7sylc 60 1  |-  ( ph  ->  ( A. x ps 
->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073
This theorem is referenced by:  spcdv  3154  spcimedv  3155  rspcimdv  3173  mrieqv2d  14688  mreexexlemd  14693
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