Theorem nfafv 48011

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6892. To prove a deduction version of this analogous to nffvd 6894 is not easily possible because a deduction version of nfdfat 48002 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 48007	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		nfafv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 48002	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5	2, 3	nffv 6892	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴)
6		nfcv 2924	. . 3 ⊢ Ⅎ𝑥V
7	4, 5, 6	nfif 4516	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
8	1, 7	nfcxfr 2922	1 ⊢ Ⅎ𝑥(𝐹'''𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2909  Vcvv 3453  ifcif 4485  ‘cfv 6537   defAt wdfat 47991  '''cafv 47992

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47960  df-dfat 47994  df-afv 47995

This theorem is used by:  csbafv12g  48012  nfaov  48054