Theorem nfif 4531

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 4530	. 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵))
8	7	mptru 1547	1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 2883  ifcif 4500

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-if 4501

This theorem is referenced by:  csbif  4558  nfop  4865  nfrdg  8428  boxcutc  8955  nfoi  9528  nfsum1  15706  nfsum  15707  summolem2a  15731  zsum  15734  sumss  15740  sumss2  15742  fsumcvg2  15743  nfcprod  15925  cbvprod  15929  prodmolem2a  15950  zprod  15953  fprod  15957  fprodntriv  15958  prodss  15963  pcmpt  16912  pcmptdvds  16914  gsummpt1n0  19946  madugsum  22581  mbfpos  25604  mbfposb  25606  i1fposd  25660  isibl2  25719  nfitg  25728  cbvitg  25729  itgss3  25768  itgcn  25798  limcmpt  25836  rlimcnp2  26928  nosupbnd2  27680  noinfbnd2  27695  chirred  32376  cdleme31sn  40399  cdleme32d  40463  cdleme32f  40465  refsum2cn  45062  ssfiunibd  45338  uzub  45458  limsupubuz  45742  icccncfext  45916  fourierdlem86  46221  vonicc  46714  nfafv  47165  nfafv2  47247