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  11916  nfsum  11917  sumrbdclem  11937  summodclem2a  11941  zsumdc  11944  fsum3  11947  isumss  11951  isumss2  11953  fsum3cvg2  11954  nfcprod1  12114  nfcprod  12115  cbvprod  12118  prodrbdclem  12131  prodmodclem2a  12136  zproddc  12139  fprodseq  12143  fprodntrivap  12144  prodssdc  12149  pcmpt  12915  pcmptdvds  12917