ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfifd Unicode version

Theorem nfifd 3465
Description: Deduction version of nfif 3466. (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 df-if 3441 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  A  /\  ps )  \/  (
y  e.  B  /\  -.  ps ) ) }
2 nfv 1491 . . 3  |-  F/ y
ph
3 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
43nfcrd 2269 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfand 1530 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
7 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
87nfcrd 2269 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
95nfnd 1618 . . . . 5  |-  ( ph  ->  F/ x  -.  ps )
108, 9nfand 1530 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  /\  -.  ps ) )
116, 10nford 1529 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  A  /\  ps )  \/  (
y  e.  B  /\  -.  ps ) ) )
122, 11nfabd 2274 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  A  /\  ps )  \/  ( y  e.  B  /\  -.  ps ) ) } )
131, 12nfcxfrd 2253 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680   F/wnf 1419    e. wcel 1463   {cab 2101   F/_wnfc 2242   ifcif 3440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-if 3441
This theorem is referenced by:  nfif  3466
  Copyright terms: Public domain W3C validator