ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsbcd Unicode version

Theorem nfsbcd 2806
Description: Deduction version of nfsbc 2807. (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 2788 . 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 2212 . . 3  |-  ( ph  -> 
F/_ x { y  |  ps } )
62, 5nfeld 2209 . 2  |-  ( ph  ->  F/ x  A  e. 
{ y  |  ps } )
71, 6nfxfrd 1380 1  |-  ( ph  ->  F/ x [. A  /  y ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1365    e. wcel 1409   {cab 2042   F/_wnfc 2181   [.wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-sbc 2788
This theorem is referenced by:  nfsbc  2807  nfcsbd  2911  sbcnestgf  2925
  Copyright terms: Public domain W3C validator