Theorem ifeq1dadc 3601

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 3588	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
3		iffalse 3579	. . . 4 |- ( -. ps -> if ( ps , A , C ) = C )
4		iffalse 3579	. . . 4 |- ( -. ps -> if ( ps , B , C ) = C )
5	3, 4	eqtr4d 2241	. . 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 838	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	2, 6, 9	mpjaodan 800	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   ifcif 3571

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-if 3572

This theorem is referenced by:  sumeq2  11670  isumss  11702  prodeq2  11868  lgsval2lem  15487  lgsval4lem  15488  lgsneg  15501  lgsmod  15503  lgsdilem2  15513