MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcsb1d Unicode version

Theorem nfcsb1d 3111
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 3082 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfv 1605 . . 3  |-  F/ y
ph
3 nfcsb1d.1 . . . 4  |-  ( ph  -> 
F/_ x A )
43nfsbc1d 3008 . . 3  |-  ( ph  ->  F/ x [. A  /  x ]. y  e.  B )
52, 4nfabd 2438 . 2  |-  ( ph  -> 
F/_ x { y  |  [. A  /  x ]. y  e.  B } )
61, 5nfcxfrd 2417 1  |-  ( ph  -> 
F/_ x [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   {cab 2269   F/_wnfc 2406   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  nfcsb1  3112
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
  Copyright terms: Public domain W3C validator