Theorem nfif 3554

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 3553	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1357	1 |- F/_ x if ( ph , A , B )

Syntax hints:   T. wtru 1349   F/wnf 1453   F/_wnfc 2299   ifcif 3526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-if 3527

This theorem is referenced by:  nfsum1  11319  nfsum  11320  sumrbdclem  11340  summodclem2a  11344  zsumdc  11347  fsum3  11350  isumss  11354  isumss2  11356  fsum3cvg2  11357  nfcprod1  11517  nfcprod  11518  cbvprod  11521  prodrbdclem  11534  prodmodclem2a  11539  zproddc  11542  fprodseq  11546  fprodntrivap  11547  prodssdc  11552  pcmpt  12295  pcmptdvds  12297