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Theorem nfsbcd 3182
Description: Deduction version of nfsbc 3183. (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 3163 . 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 2592 . . 3  |-  ( ph  -> 
F/_ x { y  |  ps } )
62, 5nfeld 2588 . 2  |-  ( ph  ->  F/ x  A  e. 
{ y  |  ps } )
71, 6nfxfrd 1581 1  |-  ( ph  ->  F/ x [. A  /  y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1554    e. wcel 1726   {cab 2423   F/_wnfc 2560   [.wsbc 3162
This theorem is referenced by:  nfsbc  3183  nfcsbd  3285  sbcnestgf  3299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-sbc 3163
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