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Theorem nfif 3706
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 11 . . 3  |-  (  T. 
->  F/ x ph )
3 nfif.2 . . . 4  |-  F/_ x A
43a1i 11 . . 3  |-  (  T. 
->  F/_ x A )
5 nfif.3 . . . 4  |-  F/_ x B
65a1i 11 . . 3  |-  (  T. 
->  F/_ x B )
72, 4, 6nfifd 3705 . 2  |-  (  T. 
->  F/_ x if (
ph ,  A ,  B ) )
87trud 1329 1  |-  F/_ x if ( ph ,  A ,  B )
Colors of variables: wff set class
Syntax hints:    T. wtru 1322   F/wnf 1550   F/_wnfc 2510   ifcif 3682
This theorem is referenced by:  csbifg  3710  nfop  3942  nfriota1  6493  nfrdg  6608  boxcutc  7041  nfoi  7416  nfsum  12412  cbvsum  12416  summolem2a  12436  zsum  12439  fsum  12441  sumss  12445  sumss2  12447  fsumcvg2  12448  pcmpt  13188  pcmptdvds  13190  mbfpos  19410  mbfposb  19412  i1fposd  19466  isibl2  19525  nfitg  19533  cbvitg  19534  itgss3  19573  itgcn  19601  limcmpt  19637  rlimcnp2  20672  chirred  23746  nfcprod  25016  cbvprod  25020  prodmolem2a  25039  zprod  25042  fprod  25046  fprodntriv  25047  prodss  25052  itgaddnclem2  25964  refsum2cn  27377  nfafv  27669  cdleme31sn  30494  cdleme32d  30558  cdleme32f  30560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-if 3683
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