Theorem 2if2 4136

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 )

Assertion

Ref	Expression
2if2	|- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )

Proof of Theorem 2if2

Step	Hyp	Ref	Expression
1		2if2.1	. . 3 |- ( ( ph /\ ps ) -> D = A )
2		iftrue 4092	. . . 4 |- ( ps -> if ( ps , A , if ( th , B , C ) ) = A )
3	2	adantl 482	. . 3 |- ( ( ph /\ ps ) -> if ( ps , A , if ( th , B , C ) ) = A )
4	1, 3	eqtr4d 2659	. 2 |- ( ( ph /\ ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
5		2if2.2	. . . . . 6 |- ( ( ph /\ -. ps /\ th ) -> D = B )
6	5	3expa 1265	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = B )
7		iftrue 4092	. . . . . 6 |- ( th -> if ( th , B , C ) = B )
8	7	adantl 482	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ th ) -> if ( th , B , C ) = B )
9	6, 8	eqtr4d 2659	. . . 4 |- ( ( ( ph /\ -. ps ) /\ th ) -> D = if ( th , B , C ) )
10		2if2.3	. . . . . 6 |- ( ( ph /\ -. ps /\ -. th ) -> D = C )
11	10	3expa 1265	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = C )
12		iffalse 4095	. . . . . . 7 |- ( -. th -> if ( th , B , C ) = C )
13	12	eqcomd 2628	. . . . . 6 |- ( -. th -> C = if ( th , B , C ) )
14	13	adantl 482	. . . . 5 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> C = if ( th , B , C ) )
15	11, 14	eqtrd 2656	. . . 4 |- ( ( ( ph /\ -. ps ) /\ -. th ) -> D = if ( th , B , C ) )
16	9, 15	pm2.61dan 832	. . 3 |- ( ( ph /\ -. ps ) -> D = if ( th , B , C ) )
17		iffalse 4095	. . . 4 |- ( -. ps -> if ( ps , A , if ( th , B , C ) ) = if ( th , B , C ) )
18	17	adantl 482	. . 3 |- ( ( ph /\ -. ps ) -> if ( ps , A , if ( th , B , C ) ) = if ( th , B , C ) )
19	16, 18	eqtr4d 2659	. 2 |- ( ( ph /\ -. ps ) -> D = if ( ps , A , if ( th , B , C ) ) )
20	4, 19	pm2.61dan 832	1 |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  swrdccat3  13492  swrdccat  13493  swrdccat3a  13494  swrdccat3b  13496  pfxccat3  41426