Theorem nfif 3632

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

Step	Hyp	Ref	Expression
1		nfif.1	. . . 4 |- F/ x ph
2	1	a1i 9	. . 3 |- ( T. -> F/ x ph )
3		nfif.2	. . . 4 |- F/_ x A
4	3	a1i 9	. . 3 |- ( T. -> F/_ x A )
5		nfif.3	. . . 4 |- F/_ x B
6	5	a1i 9	. . 3 |- ( T. -> F/_ x B )
7	2, 4, 6	nfifd 3631	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1404	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1396   F/wnf 1506   F/_wnfc 2359   ifcif 3603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-if 3604

This theorem is referenced by:  nfsum1  11907  nfsum  11908  sumrbdclem  11928  summodclem2a  11932  zsumdc  11935  fsum3  11938  isumss  11942  isumss2  11944  fsum3cvg2  11945  nfcprod1  12105  nfcprod  12106  cbvprod  12109  prodrbdclem  12122  prodmodclem2a  12127  zproddc  12130  fprodseq  12134  fprodntrivap  12135  prodssdc  12140  pcmpt  12906  pcmptdvds  12908