Theorem nfif 3655

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 3654	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1407	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1399   F/wnf 1509   F/_wnfc 2373   ifcif 3624

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-if 3625

This theorem is referenced by:  nfsum1  12066  nfsum  12067  sumrbdclem  12088  summodclem2a  12092  zsumdc  12095  fsum3  12098  isumss  12102  isumss2  12104  fsum3cvg2  12105  nfcprod1  12265  nfcprod  12266  cbvprod  12269  prodrbdclem  12282  prodmodclem2a  12287  zproddc  12290  fprodseq  12294  fprodntrivap  12295  prodssdc  12300  pcmpt  13066  pcmptdvds  13068