Theorem fvifdc 5537

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 5515	. 2 |- ( if ( ph , A , B ) = A -> ( F ` if ( ph , A , B ) ) = ( F ` A ) )
2		fveq2 5515	. 2 |- ( if ( ph , A , B ) = B -> ( F ` if ( ph , A , B ) ) = ( F ` B ) )
3	1, 2	ifsbdc 3546	1 |- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Syntax hints:   -> wi 4  DECID wdc 834   = wceq 1353   ifcif 3534   ` cfv 5216

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-if 3535  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224

This theorem is referenced by: (None)