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Theorem nfsbcd 3290
Description: Deduction version of nfsbc 3291. (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 3270 . 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 2614 . . 3  |-  ( ph  -> 
F/_ x { y  |  ps } )
62, 5nfeld 2602 . 2  |-  ( ph  ->  F/ x  A  e. 
{ y  |  ps } )
71, 6nfxfrd 1699 1  |-  ( ph  ->  F/ x [. A  /  y ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1669    e. wcel 1889   {cab 2439   F/_wnfc 2581   [.wsbc 3269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-sbc 3270
This theorem is referenced by:  nfsbc  3291  nfcsbd  3382  sbcnestgf  3786
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