Theorem nfif 3608

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 3607	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1382	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1374   F/wnf 1484   F/_wnfc 2337   ifcif 3579

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-if 3580

This theorem is referenced by:  nfsum1  11782  nfsum  11783  sumrbdclem  11803  summodclem2a  11807  zsumdc  11810  fsum3  11813  isumss  11817  isumss2  11819  fsum3cvg2  11820  nfcprod1  11980  nfcprod  11981  cbvprod  11984  prodrbdclem  11997  prodmodclem2a  12002  zproddc  12005  fprodseq  12009  fprodntrivap  12010  prodssdc  12015  pcmpt  12781  pcmptdvds  12783