Theorem fvifdc 5508

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 5486	. 2 |- ( if ( ph , A , B ) = A -> ( F ` if ( ph , A , B ) ) = ( F ` A ) )
2		fveq2 5486	. 2 |- ( if ( ph , A , B ) = B -> ( F ` if ( ph , A , B ) ) = ( F ` B ) )
3	1, 2	ifsbdc 3532	1 |- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Syntax hints:   -> wi 4  DECID wdc 824   = wceq 1343   ifcif 3520   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-if 3521  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196

This theorem is referenced by: (None)