Theorem nfafv 47419

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6843. To prove a deduction version of this analogous to nffvd 6845 is not easily possible because a deduction version of nfdfat 47410 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 47415	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		nfafv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfafv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfdfat 47410	. . 3 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴
5	2, 3	nffv 6843	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝐴)
6		nfcv 2897	. . 3 ⊢ Ⅎ𝑥V
7	4, 5, 6	nfif 4509	. 2 ⊢ Ⅎ𝑥if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
8	1, 7	nfcxfr 2895	1 ⊢ Ⅎ𝑥(𝐹'''𝐴)

Syntax hints:  Ⅎwnfc 2882  Vcvv 3439  ifcif 4478  ‘cfv 6491   defAt wdfat 47399  '''cafv 47400

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-res 5635  df-iota 6447  df-fun 6493  df-fv 6499  df-aiota 47368  df-dfat 47402  df-afv 47403

This theorem is referenced by:  csbafv12g  47420  nfaov  47462