Theorem nfif 3634

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 3633	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1406	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1398   F/wnf 1508   F/_wnfc 2361   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-if 3606

This theorem is referenced by:  nfsum1  11937  nfsum  11938  sumrbdclem  11959  summodclem2a  11963  zsumdc  11966  fsum3  11969  isumss  11973  isumss2  11975  fsum3cvg2  11976  nfcprod1  12136  nfcprod  12137  cbvprod  12140  prodrbdclem  12153  prodmodclem2a  12158  zproddc  12161  fprodseq  12165  fprodntrivap  12166  prodssdc  12171  pcmpt  12937  pcmptdvds  12939