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Theorem nfafv 38085
Description: Bound-variable hypothesis builder for function value, analogous to nffv 5879. To prove a deduction version of this analogous to nffvd 5881 is not easily possible because a deduction version of nfdfat 38079 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 38081 . 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 38079 . . 3  |-  F/ x  F defAt  A
52, 3nffv 5879 . . 3  |-  F/_ x
( F `  A
)
6 nfcv 2582 . . 3  |-  F/_ x _V
74, 5, 6nfif 3935 . 2  |-  F/_ x if ( F defAt  A , 
( F `  A
) ,  _V )
81, 7nfcxfr 2580 1  |-  F/_ x
( F''' A )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2568   _Vcvv 3078   ifcif 3906   ` cfv 5592   defAt wdfat 38062  '''cafv 38063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5556  df-fun 5594  df-fv 5600  df-dfat 38065  df-afv 38066
This theorem is referenced by:  csbafv12g  38086  nfaov  38128
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