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Theorem nfcsb1d 3028
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 2999 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfv 1508 . . 3  |-  F/ y
ph
3 nfcsb1d.1 . . . 4  |-  ( ph  -> 
F/_ x A )
43nfsbc1d 2920 . . 3  |-  ( ph  ->  F/ x [. A  /  x ]. y  e.  B )
52, 4nfabd 2298 . 2  |-  ( ph  -> 
F/_ x { y  |  [. A  /  x ]. y  e.  B } )
61, 5nfcxfrd 2277 1  |-  ( ph  -> 
F/_ x [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   {cab 2123   F/_wnfc 2266   [.wsbc 2904   [_csb 2998
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-sbc 2905  df-csb 2999
This theorem is referenced by:  nfcsb1  3029
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