Theorem nfif 3586

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 3585	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1373	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1365   F/wnf 1471   F/_wnfc 2323   ifcif 3558

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-if 3559

This theorem is referenced by:  nfsum1  11502  nfsum  11503  sumrbdclem  11523  summodclem2a  11527  zsumdc  11530  fsum3  11533  isumss  11537  isumss2  11539  fsum3cvg2  11540  nfcprod1  11700  nfcprod  11701  cbvprod  11704  prodrbdclem  11717  prodmodclem2a  11722  zproddc  11725  fprodseq  11729  fprodntrivap  11730  prodssdc  11735  pcmpt  12484  pcmptdvds  12486