Theorem 2if2dc 3645

Description: Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)

Hypotheses

Ref	Expression
2if2.1	|- ( ( ph /\ ps ) -> D = A )
2if2.2	|- ( ( ph /\ -. ps /\ th ) -> D = B )
2if2.3	|- ( ( ph /\ -. ps /\ -. th ) -> D = C )
2if2dc.ps	|- ( ph -> DECID ps )
2if2dc.th	|- ( ( ph /\ -. ps ) -> DECID th )

Assertion

Ref	Expression
2if2dc	|- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )

Proof of Theorem 2if2dc

Step	Hyp	Ref	Expression
1		2if2.1	. . 3 |- ( ( ph /\ ps ) -> D = A )
2		iftrue 3610	. . . 4 |- ( ps -> if ( ps , A , if ( th , B , C ) ) = A )
3	2	adantl 277	. . 3 |- ( ( ph /\ ps ) -> if ( ps , A , if ( th , B , C ) ) = A )
4	1, 3	eqtr4d 2267	. 2 |- ( ( ph /\ ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
5		2if2.2	. . . . . 6 |- ( ( ph /\ -. ps /\ th ) -> D = B )
6	5	3expa 1229	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = B )
7		iftrue 3610	. . . . . 6 |- ( th -> if ( th , B , C ) = B )
8	7	adantl 277	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> if ( th , B , C ) = B )
9	6, 8	eqtr4d 2267	. . . 4 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = if ( th , B , C ) )
10		2if2.3	. . . . . 6 |- ( ( ph /\ -. ps /\ -. th ) -> D = C )
11	10	3expa 1229	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = C )
12		iffalse 3613	. . . . . . 7 |- ( -. th -> if ( th , B , C ) = C )
13	12	eqcomd 2237	. . . . . 6 |- ( -. th -> C = if ( th , B , C ) )
14	13	adantl 277	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> C = if ( th , B , C ) )
15	11, 14	eqtrd 2264	. . . 4 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = if ( th , B , C ) )
16		2if2dc.th	. . . . 5 |- ( ( ph /\ -. ps ) -> DECID th )
17		exmiddc 843	. . . . 5 |- (DECID th -> ( th \/ -. th ) )
18	16, 17	syl 14	. . . 4 |- ( ( ph /\ -. ps ) -> ( th \/ -. th ) )
19	9, 15, 18	mpjaodan 805	. . 3 |- ( ( ph /\ -. ps ) -> D = if ( th , B , C ) )
20		iffalse 3613	. . . 4 |- ( -. ps -> if ( ps , A , if ( th , B , C ) ) = if ( th , B , C ) )
21	20	adantl 277	. . 3 |- ( ( ph /\ -. ps ) -> if ( ps , A , if ( th , B , C ) ) = if ( th , B , C ) )
22	19, 21	eqtr4d 2267	. 2 |- ( ( ph /\ -. ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
23		2if2dc.ps	. . 3 |- ( ph -> DECID ps )
24		exmiddc 843	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
25	23, 24	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
26	4, 22, 25	mpjaodan 805	1 |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   ifcif 3605

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 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  pfxccat3  11314  swrdccat  11315  swrdccat3b  11320