Theorem fvifdc 5670

Description: Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)

Assertion

Ref	Expression
fvifdc	|- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Proof of Theorem fvifdc

Step	Hyp	Ref	Expression
1		fveq2 5648	. 2 |- ( if ( ph , A , B ) = A -> ( F ` if ( ph , A , B ) ) = ( F ` A ) )
2		fveq2 5648	. 2 |- ( if ( ph , A , B ) = B -> ( F ` if ( ph , A , B ) ) = ( F ` B ) )
3	1, 2	ifsbdc 3622	1 |- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Syntax hints:   -> wi 4  DECID wdc 842   = wceq 1398   ifcif 3607   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-if 3608  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341

This theorem is referenced by: (None)