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

Theorem nfcsbd 3391
Description: Deduction version of nfcsb 3392. (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 3375 . 2  |-  [_ A  /  y ]_ B  =  { z  |  [. A  /  y ]. z  e.  B }
2 nfv 1771 . . 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 2608 . . . 4  |-  ( ph  ->  F/ x  z  e.  B )
73, 4, 6nfsbcd 3299 . . 3  |-  ( ph  ->  F/ x [. A  /  y ]. z  e.  B )
82, 7nfabd 2622 . 2  |-  ( ph  -> 
F/_ x { z  |  [. A  / 
y ]. z  e.  B } )
91, 8nfcxfrd 2601 1  |-  ( ph  -> 
F/_ x [_ A  /  y ]_ B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1677    e. wcel 1897   {cab 2447   F/_wnfc 2589   [.wsbc 3278   [_csb 3374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-sbc 3279  df-csb 3375
This theorem is referenced by:  nfcsb  3392
  Copyright terms: Public domain W3C validator