Theorem ifeqdadc 3638

Description: Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

Hypotheses

Ref	Expression
ifeqda.1	|- ( ( ph /\ ps ) -> A = C )
ifeqda.2	|- ( ( ph /\ -. ps ) -> B = C )
ifeqdadc.dc	|- ( ph -> DECID ps )

Assertion

Ref	Expression
ifeqdadc	|- ( ph -> if ( ps , A , B ) = C )

Proof of Theorem ifeqdadc

Step	Hyp	Ref	Expression
1		iftrue 3610	. . . 4 |- ( ps -> if ( ps , A , B ) = A )
2	1	adantl 277	. . 3 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = A )
3		ifeqda.1	. . 3 |- ( ( ph /\ ps ) -> A = C )
4	2, 3	eqtrd 2264	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = C )
5		iffalse 3613	. . . 4 |- ( -. ps -> if ( ps , A , B ) = B )
6	5	adantl 277	. . 3 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = B )
7		ifeqda.2	. . 3 |- ( ( ph /\ -. ps ) -> B = C )
8	6, 7	eqtrd 2264	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = C )
9		ifeqdadc.dc	. . 3 |- ( ph -> DECID ps )
10		exmiddc 843	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
11	9, 10	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
12	4, 8, 11	mpjaodan 805	1 |- ( ph -> if ( ps , A , B ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = 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-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  ccatsymb  11178  swrdccat3blem  11319