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Theorem spcimedv 2746
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
spcimedv  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimedv.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
21ex 114 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ch  ->  ps ) ) )
32alrimiv 1830 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ch  ->  ps ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1493 . . 3  |-  F/ x ch
6 nfcv 2258 . . 3  |-  F/_ x A
75, 6spcimegft 2738 . 2  |-  ( A. x ( x  =  A  ->  ( ch  ->  ps ) )  -> 
( A  e.  B  ->  ( ch  ->  E. x ps ) ) )
83, 4, 7sylc 62 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465
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-v 2662
This theorem is referenced by:  rspcimedv  2765  fihashf1rn  10503
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