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Theorem dfafv2 32456
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 32441 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( iota x A F x ) ,  _V )
2 df-fv 5578 . . . 4  |-  ( F `
 A )  =  ( iota x A F x )
32eqcomi 2467 . . 3  |-  ( iota
x A F x )  =  ( F `
 A )
4 ifeq1 3933 . . 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 5 . 2  |-  if ( F defAt  A ,  ( iota x A F x ) ,  _V )  =  if ( F defAt  A ,  ( F `
 A ) ,  _V )
61, 5eqtri 2483 1  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   _Vcvv 3106   ifcif 3929   class class class wbr 4439   iotacio 5532   ` cfv 5570   defAt wdfat 32437  '''cafv 32438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-un 3466  df-if 3930  df-fv 5578  df-afv 32441
This theorem is referenced by:  afveq12d  32457  nfafv  32460  afvfundmfveq  32462  afvnfundmuv  32463  afvpcfv0  32470
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