Theorem nfafv 47525

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6854. To prove a deduction version of this analogous to nffvd 6856 is not easily possible because a deduction version of nfdfat 47516 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 47521	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		nfafv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 47516	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5	2, 3	nffv 6854	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴)
6		nfcv 2899	. . 3 ⊢ Ⅎ𝑥V
7	4, 5, 6	nfif 4512	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
8	1, 7	nfcxfr 2897	1 ⊢ Ⅎ𝑥(𝐹'''𝐴)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3442  ifcif 4481  ‘cfv 6502   defAt wdfat 47505  '''cafv 47506

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6458  df-fun 6504  df-fv 6510  df-aiota 47474  df-dfat 47508  df-afv 47509

This theorem is referenced by:  csbafv12g  47526  nfaov  47568