Theorem ifsbdc 3650

Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)

Hypotheses

Ref	Expression
ifsbdc.1	|- ( if ( ph , A , B ) = A -> C = D )
ifsbdc.2	|- ( if ( ph , A , B ) = B -> C = E )

Assertion

Ref	Expression
ifsbdc	|- (DECID ph -> C = if ( ph , D , E ) )

Proof of Theorem ifsbdc

Step	Hyp	Ref	Expression
1		exmiddc 848	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3642	. . . . 5 |- ( ph -> if ( ph , A , B ) = A )
3		ifsbdc.1	. . . . 5 |- ( if ( ph , A , B ) = A -> C = D )
4	2, 3	syl 14	. . . 4 |- ( ph -> C = D )
5		iftrue 3642	. . . 4 |- ( ph -> if ( ph , D , E ) = D )
6	4, 5	eqtr4d 2274	. . 3 |- ( ph -> C = if ( ph , D , E ) )
7		iffalse 3645	. . . . 5 |- ( -. ph -> if ( ph , A , B ) = B )
8		ifsbdc.2	. . . . 5 |- ( if ( ph , A , B ) = B -> C = E )
9	7, 8	syl 14	. . . 4 |- ( -. ph -> C = E )
10		iffalse 3645	. . . 4 |- ( -. ph -> if ( ph , D , E ) = E )
11	9, 10	eqtr4d 2274	. . 3 |- ( -. ph -> C = if ( ph , D , E ) )
12	6, 11	jaoi 728	. 2 |- ( ( ph \/ -. ph ) -> C = if ( ph , D , E ) )
13	1, 12	syl 14	1 |- (DECID ph -> C = if ( ph , D , E ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846   = wceq 1402   ifcif 3635

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 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3636

This theorem is referenced by:  fvifdc  5712  lgsneg  16057  lgsdilem  16060