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Theorem dfafv2 27658
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 27636 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( iota x A F x ) ,  _V )
2 df-fv 5395 . . . 4  |-  ( F `
 A )  =  ( iota x A F x )
32eqcomi 2384 . . 3  |-  ( iota
x A F x )  =  ( F `
 A )
4 ifeq1 3679 . . 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 2400 1  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2892   ifcif 3675   class class class wbr 4146   iotacio 5349   ` cfv 5387   defAt wdfat 27632  '''cafv 27633
This theorem is referenced by:  afveq12d  27659  nfafv  27662  afvfundmfveq  27664  afvnfundmuv  27665  afvpcfv0  27672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-un 3261  df-if 3676  df-fv 5395  df-afv 27636
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