Theorem ifeqdadc 3602

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 3575	. . . 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 2237	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = C )
5		iffalse 3578	. . . 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 2237	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = C )
9		ifeqdadc.dc	. . 3 |- ( ph -> DECID ps )
10		exmiddc 837	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
11	9, 10	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
12	4, 8, 11	mpjaodan 799	1 |- ( ph -> if ( ps , A , B ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1372   ifcif 3570

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-dc 836  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-if 3571

This theorem is referenced by:  ccatsymb  11033