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Theorem nfcsb1d 3100
Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
Hypothesis
Ref Expression
nfcsb1d.1  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfcsb1d  |-  ( ph  -> 
F/_ x [_ A  /  x ]_ B )

Proof of Theorem nfcsb1d
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3070 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfv 1538 . . 3  |-  F/ y
ph
3 nfcsb1d.1 . . . 4  |-  ( ph  -> 
F/_ x A )
43nfsbc1d 2991 . . 3  |-  ( ph  ->  F/ x [. A  /  x ]. y  e.  B )
52, 4nfabd 2349 . 2  |-  ( ph  -> 
F/_ x { y  |  [. A  /  x ]. y  e.  B } )
61, 5nfcxfrd 2327 1  |-  ( ph  -> 
F/_ x [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158   {cab 2173   F/_wnfc 2316   [.wsbc 2974   [_csb 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-sbc 2975  df-csb 3070
This theorem is referenced by:  nfcsb1  3101
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