Theorem fvifdc 5518

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 5496	. 2 |- ( if ( ph , A , B ) = A -> ( F ` if ( ph , A , B ) ) = ( F ` A ) )
2		fveq2 5496	. 2 |- ( if ( ph , A , B ) = B -> ( F ` if ( ph , A , B ) ) = ( F ` B ) )
3	1, 2	ifsbdc 3538	1 |- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )

Syntax hints:   -> wi 4  DECID wdc 829   = wceq 1348   ifcif 3526   ` cfv 5198

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-if 3527  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206

This theorem is referenced by: (None)