Theorem ifeq1dadc 3379

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 3366	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
3		iffalse 3359	. . . 4 |- ( -. ps -> if ( ps , A , C ) = C )
4		iffalse 3359	. . . 4 |- ( -. ps -> if ( ps , B , C ) = C )
5	3, 4	eqtr4d 2116	. . 3 |- ( -. ps -> if ( ps , A , C ) = if ( ps , B , C ) )
6	5	adantl 271	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
7		ifeq1dadc.dc	. . 3 |- ( ph -> DECID ps )
8		exmiddc 777	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	2, 6, 9	mpjaodan 744	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661  DECID wdc 775   = wceq 1284   ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-if 3352

This theorem is referenced by: (None)