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  12148  nfsum  12149  sumrbdclem  12170  summodclem2a  12174  zsumdc  12177  fsum3  12180  isumss  12184  isumss2  12186  fsum3cvg2  12187  nfcprod1  12347  nfcprod  12348  cbvprod  12351  prodrbdclem  12364  prodmodclem2a  12369  zproddc  12372  fprodseq  12376  fprodntrivap  12377  prodssdc  12382  pcmpt  13152  pcmptdvds  13154