Theorem nfif 3669

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 3668	. 2 |- ( T. -> F/_ x if ( ph , A , B ) )
8	7	mptru 1411	1 |- F/_ x if ( ph , A , B )

This proof depends on syntax axioms:   T. wtru 1403   F/wnf 1513   F/_wnfc 2379   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-if 3639

This theorem is used by:  nfsum1  12141  nfsum  12142  sumrbdclem  12163  summodclem2a  12167  zsumdc  12170  fsum3  12173  isumss  12177  isumss2  12179  fsum3cvg2  12180  nfcprod1  12340  nfcprod  12341  cbvprod  12344  prodrbdclem  12357  prodmodclem2a  12362  zproddc  12365  fprodseq  12369  fprodntrivap  12370  prodssdc  12375  pcmpt  13145  pcmptdvds  13147