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Theorem sbciedf 3039
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 1757 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 sbciegft 3034 . 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 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  sbcied  3040  sbc2iegf  3070  csbiebt  3130  sbcnestgf  3141  ovmpt2dxf  5989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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