Theorem ifeq1dadc 3587

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifeq1dadc.1	|- ( ( ph /\ ps ) -> A = B )
ifeq1dadc.dc	|- ( ph -> DECID ps )

Assertion

Ref	Expression
ifeq1dadc	|- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Proof of Theorem ifeq1dadc

Step	Hyp	Ref	Expression
1		ifeq1dadc.1	. . 3 |- ( ( ph /\ ps ) -> A = B )
2	1	ifeq1d 3574	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
3		iffalse 3565	. . . 4 |- ( -. ps -> if ( ps , A , C ) = C )
4		iffalse 3565	. . . 4 |- ( -. ps -> if ( ps , B , C ) = C )
5	3, 4	eqtr4d 2229	. . 3 |- ( -. ps -> if ( ps , A , C ) = if ( ps , B , C ) )
6	5	adantl 277	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
7		ifeq1dadc.dc	. . 3 |- ( ph -> DECID ps )
8		exmiddc 837	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	2, 6, 9	mpjaodan 799	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   ifcif 3557

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3157  df-if 3558

This theorem is referenced by:  sumeq2  11502  isumss  11534  prodeq2  11700  lgsval2lem  15126  lgsval4lem  15127  lgsneg  15140  lgsmod  15142  lgsdilem2  15152