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Theorem dfafv2 28100
Description: Alternative definition of  ( F''' A ) using  ( F `  A ) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
dfafv2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )

Proof of Theorem dfafv2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-afv 28078 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( iota x A F x ) ,  _V )
2 df-fv 5279 . . . 4  |-  ( F `
 A )  =  ( iota x A F x )
32eqcomi 2300 . . 3  |-  ( iota
x A F x )  =  ( F `
 A )
4 ifeq1 3582 . . 3  |-  ( ( iota x A F x )  =  ( F `  A )  ->  if ( F defAt 
A ,  ( iota
x A F x ) ,  _V )  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
53, 4ax-mp 8 . 2  |-  if ( F defAt  A ,  ( iota x A F x ) ,  _V )  =  if ( F defAt  A ,  ( F `
 A ) ,  _V )
61, 5eqtri 2316 1  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801   ifcif 3578   class class class wbr 4039   iotacio 5233   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  afveq12d  28101  nfafv  28104  afvfundmfveq  28106  afvnfundmuv  28107  afvpcfv0  28114
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-fv 5279  df-afv 28078
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