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Theorem dfafv2 27963
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 27942 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( iota x A F x ) ,  _V )
2 df-fv 5454 . . . 4  |-  ( F `
 A )  =  ( iota x A F x )
32eqcomi 2439 . . 3  |-  ( iota
x A F x )  =  ( F `
 A )
4 ifeq1 3735 . . 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 2455 1  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948   ifcif 3731   class class class wbr 4204   iotacio 5408   ` cfv 5446   defAt wdfat 27938  '''cafv 27939
This theorem is referenced by:  afveq12d  27964  nfafv  27967  afvfundmfveq  27969  afvnfundmuv  27970  afvpcfv0  27977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-afv 27942
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