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Theorem sbciedf 3026
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 423 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1745 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 sbciegft 3021 . 2  |-  ( ( A  e.  V  /\  F/ x ch  /\  A. x ( x  =  A  ->  ( ps  <->  ch ) ) )  -> 
( [. A  /  x ]. ps  <->  ch ) )
81, 2, 6, 7syl3anc 1182 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbcied  3027  sbc2iegf  3057  csbiebt  3117  sbcnestgf  3128  ovmpt2dxf  5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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