Theorem nfif 4556

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	⊢ Ⅎ𝑥𝜑
nfif.2	⊢ Ⅎ𝑥𝐴
nfif.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfif	⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)

Proof of Theorem nfif

Step	Hyp	Ref	Expression
1		nfif.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		nfif.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
5		nfif.3	. . . 4 ⊢ Ⅎ𝑥𝐵
6	5	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
7	2, 4, 6	nfifd 4555	. 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵))
8	7	mptru 1547	1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 2890  ifcif 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-if 4526

This theorem is referenced by:  csbif  4583  nfop  4889  nfrdg  8454  boxcutc  8981  nfoi  9554  nfsum1  15726  nfsum  15727  summolem2a  15751  zsum  15754  sumss  15760  sumss2  15762  fsumcvg2  15763  nfcprod  15945  cbvprod  15949  prodmolem2a  15970  zprod  15973  fprod  15977  fprodntriv  15978  prodss  15983  pcmpt  16930  pcmptdvds  16932  gsummpt1n0  19983  madugsum  22649  mbfpos  25686  mbfposb  25688  i1fposd  25742  isibl2  25801  nfitg  25810  cbvitg  25811  itgss3  25850  itgcn  25880  limcmpt  25918  rlimcnp2  27009  nosupbnd2  27761  noinfbnd2  27776  chirred  32414  cdleme31sn  40382  cdleme32d  40446  cdleme32f  40448  refsum2cn  45043  ssfiunibd  45321  uzub  45442  limsupubuz  45728  icccncfext  45902  fourierdlem86  46207  vonicc  46700  nfafv  47148  nfafv2  47230