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Theorem nfcsbd 3220
Description: Deduction version of nfcsb 3221. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfcsbd.1  |-  F/ y
ph
nfcsbd.2  |-  ( ph  -> 
F/_ x A )
nfcsbd.3  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfcsbd  |-  ( ph  -> 
F/_ x [_ A  /  y ]_ B
)

Proof of Theorem nfcsbd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-csb 3188 . 2  |-  [_ A  /  y ]_ B  =  { z  |  [. A  /  y ]. z  e.  B }
2 nfv 1626 . . 3  |-  F/ z
ph
3 nfcsbd.1 . . . 4  |-  F/ y
ph
4 nfcsbd.2 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfcsbd.3 . . . . 5  |-  ( ph  -> 
F/_ x B )
65nfcrd 2529 . . . 4  |-  ( ph  ->  F/ x  z  e.  B )
73, 4, 6nfsbcd 3117 . . 3  |-  ( ph  ->  F/ x [. A  /  y ]. z  e.  B )
82, 7nfabd 2535 . 2  |-  ( ph  -> 
F/_ x { z  |  [. A  / 
y ]. z  e.  B } )
91, 8nfcxfrd 2514 1  |-  ( ph  -> 
F/_ x [_ A  /  y ]_ B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1550    e. wcel 1717   {cab 2366   F/_wnfc 2503   [.wsbc 3097   [_csb 3187
This theorem is referenced by:  nfcsb  3221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-sbc 3098  df-csb 3188
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