Theorem nfafv 48128

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6883. To prove a deduction version of this analogous to nffvd 6885 is not easily possible because a deduction version of nfdfat 48119 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 48124	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		nfafv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 48119	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5	2, 3	nffv 6883	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴)
6		nfcv 2922	. . 3 ⊢ Ⅎ𝑥V
7	4, 5, 6	nfif 4512	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
8	1, 7	nfcxfr 2920	1 ⊢ Ⅎ𝑥(𝐹'''𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2907  Vcvv 3450  ifcif 4481  ‘cfv 6527   defAt wdfat 48108  '''cafv 48109

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-iota 6483  df-fun 6529  df-fv 6535  df-aiota 48077  df-dfat 48111  df-afv 48112

This theorem is used by:  csbafv12g  48129  nfaov  48171