MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsbc1d Structured version   Unicode version

Theorem nfsbc1d 3323
Description: Deduction version of nfsbc1 3324. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypothesis
Ref Expression
nfsbc1d.2  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfsbc1d  |-  ( ph  ->  F/ x [. A  /  x ]. ps )

Proof of Theorem nfsbc1d
StepHypRef Expression
1 df-sbc 3306 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
2 nfsbc1d.2 . . 3  |-  ( ph  -> 
F/_ x A )
3 nfab1 2593 . . . 4  |-  F/_ x { x  |  ps }
43a1i 11 . . 3  |-  ( ph  -> 
F/_ x { x  |  ps } )
52, 4nfeld 2599 . 2  |-  ( ph  ->  F/ x  A  e. 
{ x  |  ps } )
61, 5nfxfrd 1693 1  |-  ( ph  ->  F/ x [. A  /  x ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1663    e. wcel 1870   {cab 2414   F/_wnfc 2577   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-sbc 3306
This theorem is referenced by:  nfsbc1  3324  nfcsb1d  3415
  Copyright terms: Public domain W3C validator