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Theorem nfifd 3601
Description: Deduction version of nfif 3602. (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 3580 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) }
2 nfv 1609 . . 3  |-  F/ y
ph
3 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
43nfcrd 2445 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfimd 1773 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  ->  ps ) )
7 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
87nfcrd 2445 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
98, 5nfand 1775 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
106, 9nfimd 1773 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) )
112, 10nfabd 2451 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps )
) } )
121, 11nfcxfrd 2430 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    e. wcel 1696   {cab 2282   F/_wnfc 2419   ifcif 3578
This theorem is referenced by:  nfif  3602  nfriotad  6329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-if 3579
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