Theorem fvifdc 5529

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 5507	. 2 |- ( if ( ph , A , B ) = A -> ( F ` if ( ph , A , B ) ) = ( F ` A ) )
2		fveq2 5507	. 2 |- ( if ( ph , A , B ) = B -> ( F ` if ( ph , A , B ) ) = ( F ` B ) )
3	1, 2	ifsbdc 3544	1 |- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Syntax hints:   -> wi 4  DECID wdc 834   = wceq 1353   ifcif 3532   ` cfv 5208

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-if 3533  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216

This theorem is referenced by: (None)