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Theorem sbciedf 3013
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 115 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1533 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 sbciegft 3008 . 2  |-  ( ( A  e.  V  /\  F/ x ch  /\  A. x ( x  =  A  ->  ( ps  <->  ch ) ) )  -> 
( [. A  /  x ]. ps  <->  ch ) )
81, 2, 6, 7syl3anc 1249 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364   F/wnf 1471    e. wcel 2160   [.wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by:  sbcied  3014  sbc2iegf  3048  csbiebt  3111  sbcnestgf  3123  ovmpodxf  6023
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