Theorem nfif 3599

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 3598	. 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 1483   F/_wnfc 2335   ifcif 3571

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-if 3572

This theorem is referenced by:  nfsum1  11700  nfsum  11701  sumrbdclem  11721  summodclem2a  11725  zsumdc  11728  fsum3  11731  isumss  11735  isumss2  11737  fsum3cvg2  11738  nfcprod1  11898  nfcprod  11899  cbvprod  11902  prodrbdclem  11915  prodmodclem2a  11920  zproddc  11923  fprodseq  11927  fprodntrivap  11928  prodssdc  11933  pcmpt  12699  pcmptdvds  12701