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

Theorem nfifd 3920
Description: Deduction version of nfif 3921. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
nfifd.2  |-  ( ph  ->  F/ x ps )
nfifd.3  |-  ( ph  -> 
F/_ x A )
nfifd.4  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
nfifd  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )

Proof of Theorem nfifd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfif2 3894 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) }
2 nfv 1771 . . 3  |-  F/ y
ph
3 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
43nfcrd 2608 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfimd 2010 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  ->  ps ) )
7 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
87nfcrd 2608 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
98, 5nfand 2018 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
106, 9nfimd 2010 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) )
112, 10nfabd 2622 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps )
) } )
121, 11nfcxfrd 2601 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   F/wnf 1677    e. wcel 1897   {cab 2447   F/_wnfc 2589   ifcif 3892
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-or 376  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-if 3893
This theorem is referenced by:  nfif  3921
  Copyright terms: Public domain W3C validator