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Theorem nfcsb1d 3283
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 3254 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfv 1630 . . 3  |-  F/ y
ph
3 nfcsb1d.1 . . . 4  |-  ( ph  -> 
F/_ x A )
43nfsbc1d 3180 . . 3  |-  ( ph  ->  F/ x [. A  /  x ]. y  e.  B )
52, 4nfabd 2593 . 2  |-  ( ph  -> 
F/_ x { y  |  [. A  /  x ]. y  e.  B } )
61, 5nfcxfrd 2572 1  |-  ( ph  -> 
F/_ x [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   {cab 2424   F/_wnfc 2561   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  nfcsb1  3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-sbc 3164  df-csb 3254
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