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Theorem nfsbcd 2997
Description: Deduction version of nfsbc 2998. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfsbcd.1  |-  F/ y
ph
nfsbcd.2  |-  ( ph  -> 
F/_ x A )
nfsbcd.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfsbcd  |-  ( ph  ->  F/ x [. A  /  y ]. ps )

Proof of Theorem nfsbcd
StepHypRef Expression
1 df-sbc 2978 . 2  |-  ( [. A  /  y ]. ps  <->  A  e.  { y  |  ps } )
2 nfsbcd.2 . . 3  |-  ( ph  -> 
F/_ x A )
3 nfsbcd.1 . . . 4  |-  F/ y
ph
4 nfsbcd.3 . . . 4  |-  ( ph  ->  F/ x ps )
53, 4nfabd 2352 . . 3  |-  ( ph  -> 
F/_ x { y  |  ps } )
62, 5nfeld 2348 . 2  |-  ( ph  ->  F/ x  A  e. 
{ y  |  ps } )
71, 6nfxfrd 1486 1  |-  ( ph  ->  F/ x [. A  /  y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1471    e. wcel 2160   {cab 2175   F/_wnfc 2319   [.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-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-sbc 2978
This theorem is referenced by:  nfsbc  2998  nfcsbd  3107  sbcnestgf  3123
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