Theorem eqifdc 3639

Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)

Assertion

Ref	Expression
eqifdc	|- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )

Proof of Theorem eqifdc

Step	Hyp	Ref	Expression
1		exmiddc 841	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
2		simpr 110	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ph )
3		simpl 109	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = if ( ph , B , C ) )
4	2	iftrued 3609	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> if ( ph , B , C ) = B )
5	3, 4	eqtrd 2262	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = B )
6	2, 5	jca 306	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ( ph /\ A = B ) )
7	6	ex 115	. . . 4 |- ( A = if ( ph , B , C ) -> ( ph -> ( ph /\ A = B ) ) )
8		simpr 110	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> -. ph )
9		simpl 109	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = if ( ph , B , C ) )
10	8	iffalsed 3612	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> if ( ph , B , C ) = C )
11	9, 10	eqtrd 2262	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = C )
12	8, 11	jca 306	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> ( -. ph /\ A = C ) )
13	12	ex 115	. . . 4 |- ( A = if ( ph , B , C ) -> ( -. ph -> ( -. ph /\ A = C ) ) )
14	7, 13	orim12d 791	. . 3 |- ( A = if ( ph , B , C ) -> ( ( ph \/ -. ph ) -> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
15	1, 14	syl5com 29	. 2 |- (DECID ph -> ( A = if ( ph , B , C ) -> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
16		simpr 110	. . . 4 |- ( ( ph /\ A = B ) -> A = B )
17		simpl 109	. . . . 5 |- ( ( ph /\ A = B ) -> ph )
18	17	iftrued 3609	. . . 4 |- ( ( ph /\ A = B ) -> if ( ph , B , C ) = B )
19	16, 18	eqtr4d 2265	. . 3 |- ( ( ph /\ A = B ) -> A = if ( ph , B , C ) )
20		simpr 110	. . . 4 |- ( ( -. ph /\ A = C ) -> A = C )
21		simpl 109	. . . . 5 |- ( ( -. ph /\ A = C ) -> -. ph )
22	21	iffalsed 3612	. . . 4 |- ( ( -. ph /\ A = C ) -> if ( ph , B , C ) = C )
23	20, 22	eqtr4d 2265	. . 3 |- ( ( -. ph /\ A = C ) -> A = if ( ph , B , C ) )
24	19, 23	jaoi 721	. 2 |- ( ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) -> A = if ( ph , B , C ) )
25	15, 24	impbid1 142	1 |- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   ifcif 3602

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 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  ifnebibdc  3648  fodjum  7313  nninfwlporlemd  7339  xrmaxiflemcom  11760  gsumfzval  13424  subctctexmid  16366