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Theorem nfcsbd 3114
Description: Deduction version of nfcsb 3115. (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 3082 . 2  |-  [_ A  /  y ]_ B  =  { z  |  [. A  /  y ]. z  e.  B }
2 nfv 1605 . . 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 2432 . . . 4  |-  ( ph  ->  F/ x  z  e.  B )
73, 4, 6nfsbcd 3011 . . 3  |-  ( ph  ->  F/ x [. A  /  y ]. z  e.  B )
82, 7nfabd 2438 . 2  |-  ( ph  -> 
F/_ x { z  |  [. A  / 
y ]. z  e.  B } )
91, 8nfcxfrd 2417 1  |-  ( ph  -> 
F/_ x [_ A  /  y ]_ B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1531    e. wcel 1684   {cab 2269   F/_wnfc 2406   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  nfcsb  3115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-sbc 2992  df-csb 3082
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