Theorem 2if2dc 3677

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 3642	. . . 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 2274	. 2 |- ( ( ph /\ ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
5		2if2.2	. . . . . 6 |- ( ( ph /\ -. ps /\ th ) -> D = B )
6	5	3expa 1234	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = B )
7		iftrue 3642	. . . . . 6 |- ( th -> if ( th , B , C ) = B )
8	7	adantl 277	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> if ( th , B , C ) = B )
9	6, 8	eqtr4d 2274	. . . 4 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = if ( th , B , C ) )
10		2if2.3	. . . . . 6 |- ( ( ph /\ -. ps /\ -. th ) -> D = C )
11	10	3expa 1234	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = C )
12		iffalse 3645	. . . . . . 7 |- ( -. th -> if ( th , B , C ) = C )
13	12	eqcomd 2244	. . . . . 6 |- ( -. th -> C = if ( th , B , C ) )
14	13	adantl 277	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> C = if ( th , B , C ) )
15	11, 14	eqtrd 2271	. . . 4 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = if ( th , B , C ) )
16		2if2dc.th	. . . . 5 |- ( ( ph /\ -. ps ) -> DECID th )
17		exmiddc 848	. . . . 5 |- (DECID th -> ( th \/ -. th ) )
18	16, 17	syl 14	. . . 4 |- ( ( ph /\ -. ps ) -> ( th \/ -. th ) )
19	9, 15, 18	mpjaodan 810	. . 3 |- ( ( ph /\ -. ps ) -> D = if ( th , B , C ) )
20		iffalse 3645	. . . 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 2274	. 2 |- ( ( ph /\ -. ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
23		2if2dc.ps	. . 3 |- ( ph -> DECID ps )
24		exmiddc 848	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
25	23, 24	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
26	4, 22, 25	mpjaodan 810	1 |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = 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-3an 1011  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3636

This theorem is referenced by:  pfxccat3  11484  swrdccat  11485  swrdccat3b  11490