Theorem ifeq1dadc 3502

Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)

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 3489	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
3		iffalse 3482	. . . 4 |- ( -. ps -> if ( ps , A , C ) = C )
4		iffalse 3482	. . . 4 |- ( -. ps -> if ( ps , B , C ) = C )
5	3, 4	eqtr4d 2175	. . 3 |- ( -. ps -> if ( ps , A , C ) = if ( ps , B , C ) )
6	5	adantl 275	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
7		ifeq1dadc.dc	. . 3 |- ( ph -> DECID ps )
8		exmiddc 821	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	2, 6, 9	mpjaodan 787	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   = wceq 1331   ifcif 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-un 3075  df-if 3475

This theorem is referenced by:  sumeq2  11128  isumss  11160  prodeq2  11326