Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfafv Structured version   Unicode version

Theorem nfafv 27978
Description: Bound-variable hypothesis builder for function value, analogous to nffv 5737. To prove a deduction version of this analogous to nffvd 5739 is not easily possible because a deduction version of nfdfat 27972 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 27974 . 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 27972 . . 3  |-  F/ x  F defAt  A
52, 3nffv 5737 . . 3  |-  F/_ x
( F `  A
)
6 nfcv 2574 . . 3  |-  F/_ x _V
74, 5, 6nfif 3765 . 2  |-  F/_ x if ( F defAt  A , 
( F `  A
) ,  _V )
81, 7nfcxfr 2571 1  |-  F/_ x
( F''' A )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2561   _Vcvv 2958   ifcif 3741   ` cfv 5456   defAt wdfat 27949  '''cafv 27950
This theorem is referenced by:  csbafv12g  27979  nfaov  28021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-dfat 27952  df-afv 27953
  Copyright terms: Public domain W3C validator