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Theorem nfsbcd 3352
Description: Deduction version of nfsbc 3353. (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 3332 . 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 2651 . . 3  |-  ( ph  -> 
F/_ x { y  |  ps } )
62, 5nfeld 2637 . 2  |-  ( ph  ->  F/ x  A  e. 
{ y  |  ps } )
71, 6nfxfrd 1626 1  |-  ( ph  ->  F/ x [. A  /  y ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1599    e. wcel 1767   {cab 2452   F/_wnfc 2615   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-sbc 3332
This theorem is referenced by:  nfsbc  3353  nfcsbd  3452  sbcnestgf  3839
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