Theorem nfafv 45050

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6847. To prove a deduction version of this analogous to nffvd 6849 is not easily possible because a deduction version of nfdfat 45041 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
nfafv.1	⊢ Ⅎ𝑥𝐹
nfafv.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfafv	⊢ Ⅎ𝑥(𝐹'''𝐴)

Proof of Theorem nfafv

Step	Hyp	Ref	Expression
1		dfafv2 45046	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		nfafv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 45041	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5	2, 3	nffv 6847	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴)
6		nfcv 2905	. . 3 ⊢ Ⅎ𝑥V
7	4, 5, 6	nfif 4514	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
8	1, 7	nfcxfr 2903	1 ⊢ Ⅎ𝑥(𝐹'''𝐴)

Syntax hints:  Ⅎwnfc 2885  Vcvv 3443  ifcif 4484  ‘cfv 6491   defAt wdfat 45030  '''cafv 45031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-br 5104  df-opab 5166  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-res 5642  df-iota 6443  df-fun 6493  df-fv 6499  df-aiota 44999  df-dfat 45033  df-afv 45034

This theorem is referenced by:  csbafv12g  45051  nfaov  45093