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

Theorem nfafv 28104
Description: Bound-variable hypothesis builder for function value, analogous to nffv 5548. To prove a deduction version of this analogous to nffvd 5550 is not easily possible because a deduction version of nfdfat 28098 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 28100 . 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 28098 . . 3  |-  F/ x  F defAt  A
52, 3nffv 5548 . . 3  |-  F/_ x
( F `  A
)
6 nfcv 2432 . . 3  |-  F/_ x _V
74, 5, 6nfif 3602 . 2  |-  F/_ x if ( F defAt  A , 
( F `  A
) ,  _V )
81, 7nfcxfr 2429 1  |-  F/_ x
( F''' A )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2419   _Vcvv 2801   ifcif 3578   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  csbafv12g  28105  nfaov  28147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-dfat 28077  df-afv 28078
  Copyright terms: Public domain W3C validator