Theorem nfafv 41216

Description: Bound-variable hypothesis builder for function value, analogous to nffv 6198. To prove a deduction version of this analogous to nffvd 6200 is not easily possible because a deduction version of nfdfat 41210 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
nfafv.1	|- F/_ x F
nfafv.2	|- F/_ x A

Assertion

Ref	Expression
nfafv	|- F/_ x ( F''' A )

Proof of Theorem nfafv

Step	Hyp	Ref	Expression
1		dfafv2 41212	. 2 |- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )
2		nfafv.1	. . . 4 |- F/_ x F
3		nfafv.2	. . . 4 |- F/_ x A
4	2, 3	nfdfat 41210	. . 3 |- F/ x F defAt A
5	2, 3	nffv 6198	. . 3 |- F/_ x ( F ` A )
6		nfcv 2764	. . 3 |- F/_ x _V
7	4, 5, 6	nfif 4115	. 2 |- F/_ x if ( F defAt A , ( F ` A ) , _V )
8	1, 7	nfcxfr 2762	1 |- F/_ x ( F''' A )

Syntax hints:   F/_wnfc 2751   _Vcvv 3200   ifcif 4086   ` cfv 5888   defAt wdfat 41193  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  csbafv12g  41217  nfaov  41259