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Theorem nfafv 31915
Description: Bound-variable hypothesis builder for function value, analogous to nffv 5873. To prove a deduction version of this analogous to nffvd 5875 is not easily possible because a deduction version of nfdfat 31909 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
StepHypRef Expression
1 dfafv2 31911 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 nfafv.1 . . . 4  |-  F/_ x F
3 nfafv.2 . . . 4  |-  F/_ x A
42, 3nfdfat 31909 . . 3  |-  F/ x  F defAt  A
52, 3nffv 5873 . . 3  |-  F/_ x
( F `  A
)
6 nfcv 2629 . . 3  |-  F/_ x _V
74, 5, 6nfif 3968 . 2  |-  F/_ x if ( F defAt  A , 
( F `  A
) ,  _V )
81, 7nfcxfr 2627 1  |-  F/_ x
( F''' A )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2615   _Vcvv 3113   ifcif 3939   ` cfv 5588   defAt wdfat 31892  '''cafv 31893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-dfat 31895  df-afv 31896
This theorem is referenced by:  csbafv12g  31916  nfaov  31958
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