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Theorem nfifd 3383
Description: Deduction version of nfif 3384. (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 3360 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  A  /\  ps )  \/  (
y  e.  B  /\  -.  ps ) ) }
2 nfv 1437 . . 3  |-  F/ y
ph
3 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
43nfcrd 2207 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfand 1476 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
7 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
87nfcrd 2207 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
95nfnd 1563 . . . . 5  |-  ( ph  ->  F/ x  -.  ps )
108, 9nfand 1476 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  /\  -.  ps ) )
116, 10nford 1475 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  A  /\  ps )  \/  (
y  e.  B  /\  -.  ps ) ) )
122, 11nfabd 2212 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  A  /\  ps )  \/  ( y  e.  B  /\  -.  ps ) ) } )
131, 12nfcxfrd 2192 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639   F/wnf 1365    e. wcel 1409   {cab 2042   F/_wnfc 2181   ifcif 3359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-if 3360
This theorem is referenced by:  nfif  3384
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