Theorem nfif 3548

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 3547	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1352	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1344   F/wnf 1448   F/_wnfc 2295   ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-if 3521

This theorem is referenced by:  nfsum1  11297  nfsum  11298  sumrbdclem  11318  summodclem2a  11322  zsumdc  11325  fsum3  11328  isumss  11332  isumss2  11334  fsum3cvg2  11335  nfcprod1  11495  nfcprod  11496  cbvprod  11499  prodrbdclem  11512  prodmodclem2a  11517  zproddc  11520  fprodseq  11524  fprodntrivap  11525  prodssdc  11530  pcmpt  12273  pcmptdvds  12275