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Theorem nfif 3447
Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
nfif.1  |-  F/ x ph
nfif.2  |-  F/_ x A
nfif.3  |-  F/_ x B
Assertion
Ref Expression
nfif  |-  F/_ x if ( ph ,  A ,  B )

Proof of Theorem nfif
StepHypRef Expression
1 nfif.1 . . . 4  |-  F/ x ph
21a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
3 nfif.2 . . . 4  |-  F/_ x A
43a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
5 nfif.3 . . . 4  |-  F/_ x B
65a1i 9 . . 3  |-  ( T. 
->  F/_ x B )
72, 4, 6nfifd 3446 . 2  |-  ( T. 
->  F/_ x if (
ph ,  A ,  B ) )
87mptru 1308 1  |-  F/_ x if ( ph ,  A ,  B )
Colors of variables: wff set class
Syntax hints:   T. wtru 1300   F/wnf 1404   F/_wnfc 2227   ifcif 3421
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-if 3422
This theorem is referenced by:  nfsum1  10964  nfsum  10965  sumrbdclem  10984  summodclem2a  10989  zsumdc  10992  fsum3  10995  isumss  10999  isumss2  11001  fsum3cvg2  11002
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