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Theorem nfifd 3588
Description: Deduction version of nfif 3589. (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 3567 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) }
2 nfv 1605 . . 3  |-  F/ y
ph
3 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
43nfcrd 2432 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfimd 1761 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  ->  ps ) )
7 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
87nfcrd 2432 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
98, 5nfand 1763 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
106, 9nfimd 1761 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) )
112, 10nfabd 2438 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps )
) } )
121, 11nfcxfrd 2417 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1531    e. wcel 1684   {cab 2269   F/_wnfc 2406   ifcif 3565
This theorem is referenced by:  nfif  3589  nfriotad  6313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-if 3566
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