Theorem nfif 3589

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 3588	. 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 1474   F/_wnfc 2326   ifcif 3561

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-if 3562

This theorem is referenced by:  nfsum1  11521  nfsum  11522  sumrbdclem  11542  summodclem2a  11546  zsumdc  11549  fsum3  11552  isumss  11556  isumss2  11558  fsum3cvg2  11559  nfcprod1  11719  nfcprod  11720  cbvprod  11723  prodrbdclem  11736  prodmodclem2a  11741  zproddc  11744  fprodseq  11748  fprodntrivap  11749  prodssdc  11754  pcmpt  12512  pcmptdvds  12514