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Theorem nfifd 3764
Description: Deduction version of nfif 3765. (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 3743 . 2  |-  if ( ps ,  A ,  B )  =  {
y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) }
2 nfv 1630 . . 3  |-  F/ y
ph
3 nfifd.4 . . . . . 6  |-  ( ph  -> 
F/_ x B )
43nfcrd 2587 . . . . 5  |-  ( ph  ->  F/ x  y  e.  B )
5 nfifd.2 . . . . 5  |-  ( ph  ->  F/ x ps )
64, 5nfimd 1828 . . . 4  |-  ( ph  ->  F/ x ( y  e.  B  ->  ps ) )
7 nfifd.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
87nfcrd 2587 . . . . 5  |-  ( ph  ->  F/ x  y  e.  A )
98, 5nfand 1844 . . . 4  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
106, 9nfimd 1828 . . 3  |-  ( ph  ->  F/ x ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps ) ) )
112, 10nfabd 2593 . 2  |-  ( ph  -> 
F/_ x { y  |  ( ( y  e.  B  ->  ps )  ->  ( y  e.  A  /\  ps )
) } )
121, 11nfcxfrd 2572 1  |-  ( ph  -> 
F/_ x if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   F/wnf 1554    e. wcel 1726   {cab 2424   F/_wnfc 2561   ifcif 3741
This theorem is referenced by:  nfif  3765  nfriotad  6560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-if 3742
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