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Theorem sbciedf 2850
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1  |-  ( ph  ->  A  e.  V )
sbcied.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
sbciedf.3  |-  F/ x ph
sbciedf.4  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
sbciedf  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    ch( x)    V( x)

Proof of Theorem sbciedf
StepHypRef Expression
1 sbcied.1 . 2  |-  ( ph  ->  A  e.  V )
2 sbciedf.4 . 2  |-  ( ph  ->  F/ x ch )
3 sbciedf.3 . . 3  |-  F/ x ph
4 sbcied.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
54ex 113 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1456 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 sbciegft 2845 . 2  |-  ( ( A  e.  V  /\  F/ x ch  /\  A. x ( x  =  A  ->  ( ps  <->  ch ) ) )  -> 
( [. A  /  x ]. ps  <->  ch ) )
81, 2, 6, 7syl3anc 1170 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   F/wnf 1390    e. wcel 1434   [.wsbc 2816
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-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  sbcied  2851  sbc2iegf  2885  csbiebt  2943  sbcnestgf  2954  ovmpt2dxf  5657
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