Theorem nfif 3647

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 3646	. 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 2371   ifcif 3616

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-if 3617

This theorem is referenced by:  nfsum1  12019  nfsum  12020  sumrbdclem  12041  summodclem2a  12045  zsumdc  12048  fsum3  12051  isumss  12055  isumss2  12057  fsum3cvg2  12058  nfcprod1  12218  nfcprod  12219  cbvprod  12222  prodrbdclem  12235  prodmodclem2a  12240  zproddc  12243  fprodseq  12247  fprodntrivap  12248  prodssdc  12253  pcmpt  13019  pcmptdvds  13021