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Theorem nfafv 27999
Description: Bound-variable hypothesis builder for function value, analogous to nffv 5532. To prove a deduction version of this analogous to nffvd 5534 is not easily possible because a deduction version of nfdfat 27993 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 27995 . 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 27993 . . 3  |-  F/ x  F defAt  A
52, 3nffv 5532 . . 3  |-  F/_ x
( F `  A
)
6 nfcv 2419 . . 3  |-  F/_ x _V
74, 5, 6nfif 3589 . 2  |-  F/_ x if ( F defAt  A , 
( F `  A
) ,  _V )
81, 7nfcxfr 2416 1  |-  F/_ x
( F''' A )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2406   _Vcvv 2788   ifcif 3565   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  csbafv12g  28000  nfaov  28039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975
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