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Theorem dfafv2 30181
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 30164 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( iota x A F x ) ,  _V )
2 df-fv 5529 . . . 4  |-  ( F `
 A )  =  ( iota x A F x )
32eqcomi 2465 . . 3  |-  ( iota
x A F x )  =  ( F `
 A )
4 ifeq1 3898 . . 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 2481 1  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3072   ifcif 3894   class class class wbr 4395   iotacio 5482   ` cfv 5521   defAt wdfat 30160  '''cafv 30161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-rab 2805  df-v 3074  df-un 3436  df-if 3895  df-fv 5529  df-afv 30164
This theorem is referenced by:  afveq12d  30182  nfafv  30185  afvfundmfveq  30187  afvnfundmuv  30188  afvpcfv0  30195
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