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Theorem nfcsbd 3040
Description: Deduction version of nfcsb 3041. (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 3007 . 2  |-  [_ A  /  y ]_ B  =  { z  |  [. A  /  y ]. z  e.  B }
2 nfv 1509 . . 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 2296 . . . 4  |-  ( ph  ->  F/ x  z  e.  B )
73, 4, 6nfsbcd 2931 . . 3  |-  ( ph  ->  F/ x [. A  /  y ]. z  e.  B )
82, 7nfabd 2301 . 2  |-  ( ph  -> 
F/_ x { z  |  [. A  / 
y ]. z  e.  B } )
91, 8nfcxfrd 2280 1  |-  ( ph  -> 
F/_ x [_ A  /  y ]_ B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1437    e. wcel 1481   {cab 2126   F/_wnfc 2269   [.wsbc 2912   [_csb 3006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-sbc 2913  df-csb 3007
This theorem is referenced by:  nfcsb  3041
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