Theorem ifsbdc 3569

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 837	. 2 |- (DECID ph -> ( ph \/ -. ph ) )
2		iftrue 3562	. . . . 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 3562	. . . 4 |- ( ph -> if ( ph , D , E ) = D )
6	4, 5	eqtr4d 2229	. . 3 |- ( ph -> C = if ( ph , D , E ) )
7		iffalse 3565	. . . . 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 3565	. . . 4 |- ( -. ph -> if ( ph , D , E ) = E )
11	9, 10	eqtr4d 2229	. . 3 |- ( -. ph -> C = if ( ph , D , E ) )
12	6, 11	jaoi 717	. 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 709  DECID wdc 835   = wceq 1364   ifcif 3557

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-if 3558

This theorem is referenced by:  fvifdc  5576  lgsneg  15140  lgsdilem  15143